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Reteaching Masters
To jump to a location in this book
1. Click a bookmark on the left.
To print a part of the book
1. Click the Print button.
2. When the Print window opens, type in a range of pages to print.
The page numbers are displayed in the bar at the bottom of the document. In the example below,“1 of 151” means that the current page is page 1 in a file of 151 pages.
NAME _________________________________________________ CLASS _______________ DATE ______________
2 Reteaching 1.1 Algebra 2
◆Skill B Graphing a linear equation
Recall If x and y are linearly related, this relationship can be expressed as a linear equationin the form y � mx � b.
◆ Example Graph the linear equation y � 2x �3.
◆ SolutionSince it takes two points to determine a line, picktwo x-values and use the equation to calculatethe corresponding y-values. Graph the orderedpairs (x, y) and draw a line. Use a third point tocheck that all three points are on the same line.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 1.3 5
◆Skill A Writing a specified linear equation in slope-intercept form
Recall The point-slope form of the equation of a line with slope m that contains(x1, y1) is y � y1 � m(x � x1).
◆ Example Write an equation in slope-intercept form for the line containing the points(1, �1) and (2, 3).
◆ SolutionFind the slope of the line; .
Use the form y � y1 � m(x � x1). Replace m with 4, x1 with 2, and y1 with 3.
y � 3 � 4(x � 2)y � 3 � 4x � 8
y � 4x � 5 This is the slope-intercept form.
The slope is 4 and the y-intercept is �5.
Notice that if you had used the point (1,�1) rather than (2, 3), then you wouldhave the following:
y � (�1) � 4(x � 1)y � 4x � 5
m �3 � (�1)
2 � 1�
41
� 4
Write an equation for the line containing each pair of points.Then state the slope and the y-intercept of each line.
1. (0, 0) and (�4, 2) 2. (6, 2) and (�2, 6) 3. (�2, �2) and (5, 5)
y � y � y �
slope: slope: slope:
y-intercept: y-intercept: y-intercept:
4. (7, 0) and (0, �4) 5. (�3, �2) and (5, �2) 6. (6, 1) and (0, �3)
y � y � y �
slope: slope: slope:
y-intercept: y-intercept: y-intercept:
Reteaching
1.3 Linear Equations in Two Variables
Write an equation in slope-intercept form for the line thatcontains the given point and is parallel to the given line. Thenwrite the equation for the line that contains the same point andis perpendicular to the given line.
NAME _________________________________________________ CLASS _______________ DATE ______________
6 Reteaching 1.3 Algebra 2
◆Skill B Writing equations for parallel or perpendicular lines
Recall Parallel lines have the same slope; perpendicular lines have slopes that are negativereciprocals of each other.
◆ Example 1Write an equation in slope-intercept form for the line that contains the point (2, 1) and is parallel to the line whose equation is y � 2x � 5.
◆ SolutionThe slope of the line with equation y � 2x � 5 is 2. Any line parallel to this linewill have a slope of 2. Use the point-slope form.
y � y1 � m(x – x1)y � 1 � 2(x – 2)y � 1 � 2x – 4
y � 2x � 3You can use a graphics calculator to check that these two lines are parallel.
◆ Example 2Write an equation in slope-intercept form for the line that contains the point (2, 1) and is perpendicular to the line whose equation is y � 2x � 5.
◆ SolutionSince the slope of the line with equation y � 2x � 5 is 2, the slope of the
perpendicular line will be the negative reciprocal of 2, which is .
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 1.4 7
◆Skill A Writing a direct variation equation
Recall An equation in the form y � kx is called a direct variation equation where k is theconstant of variation.
◆ Example The total cost of tickets to a local concert varies directly as the number of ticketsyou buy. If 4 tickets cost $72, find the constant of variation, k, and write anequation to show this direct variation. Then find the cost of 11 tickets.
◆ SolutionLet x represent the number of tickets and y represent the total cost. Then forsome value of k, y = kx.
72 � k(4)
k � 18Each ticket costs $18.An equation for the direct variation is y � 18x.To find the cost of 11 tickets, substitute 11 for x. Then find y.y � 18(11) � 198Thus, 11 tickets will cost $198.
724
� k
In Exercises 1–6, y varies directly as x. Find the constant ofvariation, and write an equation of direct variation that relatesthe two variables.
1. y � 40 when x � 5 2. y � 6 when x � 12 3. y � 15 when x � 10
k � k � k �
equation: equation: equation:
4. y � 1 when 5. y � 2 when x � �8 6. y � 3 when x � �0.2
k � k � k �
equation: equation: equation:
Use a direct variation equation to solve each problem.
7. If 5 tickets cost $65, find the cost of 8 tickets.
8. If 4 cassette tapes on sale cost $28, find the cost of 10 tapes.
9. If 6 colas cost $3.18, find the cost of 20 colas.
10. If 5 pens cost $1.75, find the cost of 18 pens.
x �34
Reteaching
1.4 Direct Variation and Proportion
Solve each proportion for the indicated variable. Check your answers.
11. 12. 13.
14. 15. 16.
Use a proportion to solve each problem.
17. If exactly 2 out of 3 students voted for candidate A, how many votes out
of the 120 cast were for candidate A?
18. If 8 gallons of gasoline cost $9.44, what is the cost for 14 gallons?
19. If Monica jogs at a constant speed and covers 2 miles in 13 minutes, how
long will it take her to jog 5 miles?
20. In the smaller of 2 similar right triangles, the hypotenuse measures 10centimeters and the shorter leg measures 3 centimeters. In the largertriangle the shorter leg measures 8 centimeters. How long is the
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 1.5 9
◆Skill A Using a scatter plot, least-squares line, and correlation coefficient to analyze data
Recall The least-squares line for a positive correlation will have a positive slope.
◆ Example Draw a scatter plot for the following data. Find an equation for the least-squaresline. Draw this line and describe the correlation.
◆ SolutionEnter the data in your calculator to find aleast-squares line of y ≈ 6.93x � 1.39 and acorrelation coefficient of r ≈ 0.994.
The graph and this value of r indicate a strongpositive correlation.
Create a scatter plot for the data in each table below. Describethe correlation. Then find an equation for the least-squares line.Draw this line on the scatter plot.
1. 2.
x
y
x
y
Reteaching
1.5 Scatter Plots and Least-Square Lines
X 2 3 4 5 6 7 8 9
Y 14 24 30 33 43 52 58 62
x
y
20
40
60
80
2 4 6 8 10
x 3 4 5 6 7 8
y 56 32 74 50 92 78
x 1 3 5 7 9 11
y 8 7 5 4 2 1
Use a graphics calculator to find the equation of the least-squares line for each set of data. Then find each value of y.Round answers to the nearest hundredth.
NAME _________________________________________________ CLASS _______________ DATE ______________
10 Reteaching 1.5 Algebra 2
◆Skill B Using a least-squares line to predict or estimate values of a variable
Recall Once you have found an equation for the least-squares line, you can use substitution to estimate or make a prediction of the value of the second variable.
◆ Example Randy used the following table to record miles he had driven and the amount of gas used.
Estimate the amount of gas needed for a trip of 275 miles and for a trip of 800 miles.
◆ SolutionUse a graphics calculator to find the equation of the least-squares line for this data.
y ≈ 0.037x � 0.2
Evaluate y ≈ 0.037x � 0.2 with x � 275. Then y ≈ 10.4.The 275-mile trip will require approximately 10.4 gallons of gas.
Evaluate y ≈ 0.037x � 0.2 with x � 800. Then y � 29.8.The 800-mile trip will require approximately 29.8 gallons of gas.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 1.6 11
◆Skill A Solving linear equations in one variable
Recall An equation is solved by using inverse operations.
◆ Example 1The total cost for a set of 4 CDs, including shipping and handling charges, is$49.50. If the shipping and handling charges are $3.50, what is the cost of each CD?
◆ Solution4n � 3.5 � 49.5 where n is the cost of 1 CD
4n � 3.5 � 3.5 � 49.5 � 3.5 Subtract 3.5 from each side of the equation.4n � 46
Divide each side by 4.n � 11.5
Each CD costs $11.50.To check your answer, show that 4(11.5) + 3.5 = 49.5.
◆ Example 2Solve 3x � 5 � 7x � 12.
◆ Solution3x � 5 � 7x � 12
3x � 5 � 3x � 7x � 12 � 3x Subtract 3x from each side of the equation.�5 � 4x � 12 Combine like terms.
�5 � 12 � 4x � 12 � 12�17 � 4x
Thus, x � , or .�414
�174
�174
� x
�174
�4x4
4n4
�464
Solve each equation.
1. 3x � 4 � 5 2.
3. 4. 1.5t � 3 � 7.5
5. 2(x � 3) � x � 3 6. 3(y � 1) � �(y � 5)
7. 2r � 1 � r � 5 � 3r 8.
9. 0.2(x � 5) � x � 5 10. 2a � (1 � a) � 11 � a
11. 12. 3(5y � 1) � 5(3y� 2) � 725
(x � 2) � x � 4
13
(c � 2) � c � 4
15 �16
a � �1
23
y � 8 � 0
Reteaching
1.6 Introduction to Solving Equations
Solve each literal equation for the indicated variable.
NAME _________________________________________________ CLASS _______________ DATE ______________
12 Reteaching 1.6 Algebra 2
◆Skill B Solving literal equations for a specified variable
Recall Many formulas are literal equations that contain two or more variables.
◆ Example 1The formula for changing Celsius to Fahrenheit temperature is . To find a formula for changing Fahrenheit to Celsius, solvethis equation for C.
◆ Solution
Subtract 32 from each side of the equation.
Multiply each side of the equation by .
Thus, the equation for changing Fahrenheit temperature to Celsius is
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 2.1 17
◆Skill A Classifying real numbers
Recall You can classify a real number as belonging to the natural numbers, wholenumbers, integers, rational numbers, or irrational numbers. A real number canbelong to more than one set of numbers.
◆ ExampleClassify �4 in as many ways as possible.
◆ Solution4 is not a natural number because natural numbers are positive whole
numbers.4 is not a whole number because whole numbers are either positive or 0.4 is an integer because integers are all the whole numbers and their opposites.4 is a rational number because it can be written as the terminating decimal �4.0.4 is a real number.
The number 4 is an integer, a rational number, and a real number.�
�
�
�
�
�
Use the diagram to classify each number in as many ways as possible by writing it in the smallest
rectangle in which it belongs. For example, is
placed in the rectangle labeled rational.
1. �8 2. 25 3. 6.8
4. 5. 6.
7. 8. 9.
10. 3π
05
��25311
�1.6513
�2
23
Reteaching
2.1 Operations With Numbers
Real Numbers
Rational
Integers
Whole
Natural
Irrational23
◆Skill B Identifying properties of real numbers
Recall The real numbers are characterized by the Commutative and Associative Propertiesof Addition and Multiplication and by the Distributive Property.
◆ ExampleTell if the statement is true or false. Justify your response.
a. b. c.
◆ Solution
a. True Commutative Property of Multiplicationb. False Subtraction is not associative.c. True Distributive Property
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 2.3 21
◆Skill A Identifying that a given relation is a function
Recall A function is a relation in which each value in the domain is paired with exactlyone value in the range.
◆ Example 1Does the table at right represent a function?
◆ SolutionYes; for each value of x there is only one value of y. Notice that the two x-values of �2 and 2 have the same y-value, 4. This is allowed in the definition of function.
◆ Example 2Does the solid-line graph shown represent a function? State the domain and range.
◆ SolutionYes; notice that any vertical line will intersect the graph in no more than 1 point.
domain: range: y � �4x � �2
State whether each relation represents a function and give thedomain and range.
NAME _________________________________________________ CLASS _______________ DATE ______________
22 Reteaching 2.3 Algebra 2
◆Skill B Writing and evaluating functions
Recall The value of depends on the value of x.
◆ Example 1Sarah uses an internet server which charges $12.50 per month plus $0.60 foreach hour over 20 hours that she uses it during the month. Write this relation infunction notation. How much will she be charged for using the service for 38hours in April?
◆ SolutionLet h � number of hours over 20. Thus, the function is as follows.
where h � 18
The charge for April will be $35.30.
◆ Example 2If , find .
◆ Solutionmeans replace x with the value and evaluate .
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 2.4 23
◆Skill A Using the four basic operations on functions to write new functions
Recall To write the sum, difference, product, or quotient of two functions, f and g, writethe sum, difference, product, or quotient of the expressions that define f and g.Then simplify.
◆ ExampleLet and . Write an expression for each function.
NAME _________________________________________________ CLASS _______________ DATE ______________
24 Reteaching 2.4 Algebra 2
◆Skill B Finding the composite of two functions
Recall To write an expression for the composite function (x), replace each x in theexpression for f with the expression defining g. Then simplify the result.
◆ ExampleLet and . Find (2) and (2). Then writeexpressions for (x) and (x).
◆ Solution(2):
Thus, (2) .
(2):
Thus, (2) .
To write expressions for (x) and (x), use the variable x instead of aparticular number.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 2.5 25
◆Skill A Finding inverses of functions
Recall The inverse of a relation is found by interchanging x and y and then solving for y.
◆ Example 1Find the inverse of the function given by {(2, 7), (5, 13), (7, 19), (9, 25)}. Tellwhether the inverse relation is a function.
◆ SolutionInterchange x and y. {(7, 2), (13, 5), (19, 7), (25, 9)}The inverse is a function because there is only one y-value for each x-value.
◆ Example 2Find the inverse, , of . The find and .
◆ SolutionReplace with y.Interchange x and y.Solve for y.
� x� x
� x �52
�52
� x � 5 � 5
�12
(2x � 5) �52
� 2(12
x �52) � 5
� f �1(2x � 5)f �1(f(x))� f (12
x �52)f(f �1(x))
y �12
x �52
x � 52
� y
x � 5 � 2yx � 2y � 5y � 2x � 5f(x)
f �1(f(x))f(f �1(x))f(x) � 2x � 5f �1
Find the inverse, , of each function. Then find and.
1. 2.
3. 4.
f �1(f(5)) �f(f �1(5)) �f �1(f(5)) �f(f �1(5)) �
f �1(x) �f �1(x) �
f(x) �2x � 1
5f(x) � 7x � 4
f �1(f(5)) �f(f �1(5)) �f �1(f(5)) �f(f �1(5)) �
f �1(x) �f �1(x) �
f(x) �12
x � 3f(x) � {(�5, �1), (0, 3), (3, 5), (5, 6)}
f �1(f(5))f(f �1(5))f �1(x)
Reteaching
2.5 Inverses of Functions
Use x-values of �2, �1, 0, 1, and 2 to find ordered pairs and grapheach function. On the same coordinate grid, sketch the inverserelation and determine whether it is a function.
NAME _________________________________________________ CLASS _______________ DATE ______________
26 Reteaching 2.5 Algebra 2
◆Skill B Using the horizontal-line test and graphing the inverse of a function
Recall The inverse of a function is also a function if and only if every horizontal line inter-sects the graph of the given function in no more than one point.
◆ Examplea. Graph the function given by . Then use the horizontal-line test
to find out if its inverse relation is also a function.b. Sketch the line and the inverse relation on the same set of axes.
◆ Solutiona. Use several values of x to find ordered pairs.
{(�2, 5), (�1, 2), (0, 1), (1, 2), (2, 5)}
Graph these points and connect them with a smooth curve.
The horizontal line drawn shows that this function does not pass the horizontal-line test.Therefore, the inverse relation is not a function.
b. Use {(5, �2), (2, �1), (1, 0), (2, 1), (5, 2)} to graph the inverse relation. Notice that the inverse is the reflection of the graph of
across the line y � x.
Also observe that the inverse relation will not pass the vertical-line test. This confirms that the inverse relation is not a function.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 2.6 27
◆Skill A Evaluating and applying rounding-up, rounding-down, and absolute valuefunctions
Recall The rounding-up function rounds a decimal value to the next highest integer.
◆ Example 1A long distance telephone company advertises that weekend calls cost $0.10 perminute. Each fraction of a minute is rounded up to the next whole minute.Write this as a rounding-up function. Then find the cost of a 23.5-minute call.
◆ Solutionwhere m � number of minutes
24 is the next highest integer after 23.5
The call will cost $2.40.
Recall The rounding-down function rounds a decimal value to the next lowest integer.
◆ Example 2Evaluate .
◆ Solution�4 is the next integer to the left of �3.7
Recall Absolute value means distance from 0.
◆ Example 3Evaluate .
◆ Solution�12.3 is at a distance 12.3 units from 0� 12.3��12.3�
��12.3�
��2.3� � 2.3
� �4[�3.7]
[�3.7]
[�2.3] � �3
� 2.40� 0.1(24)
f(23.5) � 0.123.5f(m) � 0.1m
�2.3 � �2
Evaluate.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
Write a function for each problem. Solve the problem.
13. Another phone company charges $0.12 per minute, but does not charge for the next minute unless you use the full minute. What is the charge for a 16.5 minute call?
14. A local plumber charges $50 for a house call plus $35 per hour or any fraction of an hour. What is the charge for a 3.75 hour house call?
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 3.1 31
◆Skill A Solving a system of linear equations by graphing
Recall A system of equations may be consistent or inconsistent. If it is consistent, theequations may be dependent or independent.
◆ Example Graph each system. Classify the graphs as intersecting lines, parallel lines, or the same line. Then classify the system as consistent or inconsistent. If it isconsistent, classify it as dependent or independent and find the solution.
a. b. c.
◆ SolutionThe first equation in each system is already solved for y. Solve the second equation in each system for y and graph.
→
→
→
a. intersecting lines; consistent; independent; (2, 2)b. parallel lines; inconsistent; no solutionc. same line; consistent; dependent; all ordered pairs
(x, y) such that y �12
x � 1
y �12
x � 1x � 2y � �2
y �12
x � 2y � 2 �12
x
y � �x � 4x � y � 4
�y �12
x � 1
x � 2y � �2�y �12
x � 1
y � 2 �12
x�y �12
x � 1
x � y � 4
Graph and classify each system. Then find the solution from the graph.
1. 2. 3.
x
y
Ox
y
Ox
y
O
�y � 2x � 12x � y � �3��3x � 6y � �6
x � 2y � 2�x � y � �2x � y � 6
Reteaching
3.1 Solving Systems by Graphing or Substitution
x
y
O
x + y = 4
y = x – 21
2y = x + 11
2
Use substitution to solve each system of equations. Check your solution.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 3.2 33
◆Skill A Solving a consistent and independent system of equations by elimination
Recall Two lines with different slopes represent a consistent and independent system of equations. Since these lines intersect in one point, there is one solution to the system.
◆ Example
Solve by using the elimination method.
◆ SolutionTo eliminate the y terms multiply each side of the first equation by 3.
Then multiply each side of the second equation by 2.
The system that results is shown below.
Addition Property of Equality
To find y, replace x with 3 in the first equation.
Check the ordered pair (3, 2) in the second original equation.
12 � 6 � 64(3) � 3(2) � 6
4x � 3y � 6
y � 2
2y � 43(3) � 2y � 13
3x � 2y � 13
x � 3
17x � 51
8x � 6y � 129x � 6y � 39
4x � 3y � 6 → 2(4x � 3y) � 2(6) → 8x � 6y � 12
3x � 2y � 13 → 3(3x � 2y) � 3(13) → 9x � 6y � 39
�3x � 2y � 134x � 3y � 6
Use elimination to solve each system of equations.Check your solution.
NAME _________________________________________________ CLASS _______________ DATE ______________
34 Reteaching 3.2 Algebra 2
◆Skill A Text
Recall Text
◆ ExampleSBHCT
Classify each system as consistent or inconsistent, independentor dependent. If the system is consistent, find the solution.
9. 10.
11. 12.
13. 14.
15. 16. �7x � 14y � 21x � 2y � 3�3x � 5y � 16
�x � 4y � �10
�2x � 7y � �16x � 21y � �3�4x � 3y � 15
12x � 9y � 36
�x � y � 1x � 5y � �23�2x � 7y � �4
�x � 8y � 2
�2x � y � 1810x � 5y � 90�x � 7y � 13
�x � 7y � �7
◆Skill B Classifying a system as dependent or inconsistent
Recall If the same line represents two different equations, the system is dependent.If parallel lines represent two different equations, the system is inconsistent.
◆ Example 1Classify the system as consistent or inconsistent, independent
or dependent.
◆ Solution
Since is a false statement, this system represents a pair of parallel lines. There is no solution because this system is inconsistent.
◆ Example 2Classify the system as consistent or inconsistent, independent
or dependent.
◆ Solution
Since is a true statement, these are both the same line. This system is aconsistent and dependent system.
Any solution to the first equation will also be a solution to the second equation.
Any solution to the second equation will also be a solution to the first equation.
0 � 00 � 0
�2x � 2y � 6�2x � 2y � �6�x � y � 3
�2x � 2y � �6→ �2(x � y) � 2(3)
�2x � 2y � �6→
�x � y � 3�2x � 2y � �6
0 � 70 � 7
�2x � y � 3�2x � y � 4�2x � y � 3
2x � y � �4→ �2x � y � 3
(�1)(2x � y) � (�1)(�4)→
�2x � y � 32x � y � �4
Lesson 2.7
1. f (x) � x2; reflection across the x-axis andvertical translation 5 units up
2. f (x) � |x|; horizontal translation 3 units tothe right
3. f (x) � x2; horizontal translation 2 units tothe left and vertical translation 3 units up
4. f (x) � ; reflection across the y-axis andvertical stretch by a factor of 2
5. f (x) � |x|; horizontal compression by a
factor of
6. f (x) � ; reflection across the x-axis,vertical stretch by a factor of 2, and ahorizontal translation 4 units to the left
7. g(x) � (x � 3)2 � 2
8. g(x) � �x2 � 2
9. g(x) � |x �3| � 1
10. g(x) � |2x| – 3, or g(x) = 2|x| – 3
11. g(x) �
12. g(x) � �1 �
Reteaching — Chapter 3
Lesson 3.1
1.
intersecting, consistent, independent(2, �4)
2.
same line, consistent, dependentall (x, y) such that x � 2y � 2
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 3.3 35
◆Skill A Text
Recall Text
◆ ExampleSBHCT
Reteaching
3.3 Linear Inequalities in Two Variables
◆Skill A Writing an inequality in two variables for a given graph
Recall A broken boundary line in the graph of an inequality indicates that points on theline are not part of the solution set.
◆ ExampleThe school drama club is selling tickets fortheir next production. Adult tickets cost $5and student tickets are $3. They must sell atleast $600 worth of tickets to cover expenses.The shaded part of the graph contains all theordered pairs of numbers of adult and studenttickets, (a, s), that will earn them at least $600.Determine which of the following orderedpairs are in the shaded region. Then write aninequality in two variables for the shadedportion of the graph.
(50, 100), (100, 50), (90, 90)
◆ SolutionThe point (50, 100) is not in the shaded region.Notice that ($5)(50) � ($3)(100) � $550 and $550 � $600.
The point (100, 50) is in the shaded region.($5)(100) � ($3)(50) � $600
The point (90, 90) is in the shaded region.($5)(90) � ($3)(90) � $600
The shaded region contains all the points where .5a � 3s � 600
Determine whether the point (2, 3) is in the shaded region ofeach graph. Then write an inequality in two variables for thegraph shown.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 3.6 41
◆Skill A Graphing a pair of parametric equations
Recall In a pair of parametric equations, such as and , the values ofboth x and y are dependent upon the value of the parameter t.
◆ Example(You will need a graphics calculator.)Your cat chases a bird that it sees on the lawn. The bird takes flight and itshorizontal distance increases by 10 feet per second; . At the same timeits vertical distance can be modeled by . Sketch the path of the bird for the first 5 seconds as it escapes from your cat.
◆ SolutionUse your calculator to complete the table. Plot the points and connect them with a smooth curve.
Check your graph by using your graphics calculator in parametric mode.
y(t) � 3�tx(t) � 10t
y(t) � 3�tx(t) � 10t
Graph each pair of parametric equations for .
1. 2. 3.
x
y
x
y
�x(t) �12
t2
y(t) � �t � 6�x(t) � t
y(t) � �12
t � 7�x(t) � 2ty(t) � t � 1
0 � t � 5
Reteaching
3.6 Parametric Equations
x
y
O 10 20 30 40 50
2
4
6
8
10t
0 0 0
1 10 3
2 20 4.24
3 30 5.20
4 40 6
5 50 6.71
y(t) � 3�tx(t) � 10t
x
y
Write each pair of parametric equations as a single equation in x and y.
NAME _________________________________________________ CLASS _______________ DATE ______________
42 Reteaching 3.6 Algebra 2
◆Skill B Writing a pair of parametric equations as a single equation
◆ ExampleWrite the pair of parametric equations as a single equation
involving only x and y.
◆ SolutionSolve each equation for t.
Set the two resulting expressions for t equal to each other and solve for y.
y � 2x � 132x � 12 � y � 1
x � 6 �y � 1
2
t �y � 1
2t � x � 6
y � 2t � 1x � t � 6
�x(t) � t � 6y(t) � 2t � 1
◆Skill C Using parametric equations to find the inverse of a function
Recall To find the inverse of a function, switch the x and y variables.
◆ Example 1Find the inverse for
◆ Solution
Use your graphics calculator to check that these two functions are reflectionsacross the line y � x. To graph y � x in parametric mode use and .y(t) � tx(t) � t
�x(t) � 2t � 1y(t) � t
�x(t) � ty(t) � 2t � 1
Find the inverse of each pair of parametric equations. Use agraphics calculator to check.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 4.2 45
◆Skill A Multiplying matrices
Recall When multiplying matrices, you multiply elements in a row of the first matrix by corresponding elements in a column of the second matrix. The sum of these products gives one element in the product matrix.
◆ Example
Find the matrix products AB and BA, if they exist.
◆ Solution
Multiply each element in the first row of A by the corresponding element in thefirst column of B. Then add these products.
1st row � 1st column: (2)(1) � (�1)(�4) � 6
Follow the same procedure with the first row of Aand the second column of B, and then the third column of B.
NAME _________________________________________________ CLASS _______________ DATE ______________
46 Reteaching 4.2 Algebra 2
◆Skill B Using matrices to perform transformations in the plane
Recall A matrix can be used to represent points in the plane. The first row contains the x-coordinates and the second row contains the y-coordinates of a figure.
◆ Example a. Sketch the triangle with vertices given by the matrix .
b. Multiply this matrix of vertices by the matrix to graph and describe a transformation of the original triangle.
◆ Solution
a. is graphed as A(�1, 1),
B(3, 5) and C(3, 0).
b.
The result has vertices (1, 1), (�3, 5),and (�3, 0).Triangle A'B'C' is the reflection of triangle ABC across the y-axis.
��10
01���1
135
30� � �1
1�3
5�3
0�
��11
35
30�
��10
01���1
135
30�
x
y
O
A
B
C
C’
B’
A’
x
y
O
Use the triangle given in the example above. Graph and describe the transformation that results from multiplying by each of thefollowing matrices.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 4.4 49
◆Skill A Writing a matrix equation to represent a system of equations
Recall The inverse of matrix A is written as A�1.
◆ Example Randy saved gas by only using his car to travel to and from work and the grocerystore. One week he made 5 round trips to work and 2 round trips to the store. Henoticed on the odometer that he had traveled 61.2 miles. The next week 4 roundtrips to work and 3 round trips to the store gave a total mileage of 55.4. Write amatrix equation that represents this information.
◆ SolutionA system of equations for this problem is
where w represents the number of miles in a trip to and from work, and s represents the number of miles in a trip to and from the store.
Let ,
Then the matrix equation AX � B is and represents thesame system of equations.
A � �54
23��w
s � � �61.255.4�
A � �54
23�, X � �w
s � and B � �61.255.4�
�5w � 2s � 61.24w � 3s � 55.4
Write the matrix equation that represents each system.
1. 2.
3. 4.
5. 6. �2a � b � 7a � 4c � �32b � c � 15�3x � 4y � z � 11
�2x � y � 5z � 3x � 6y � z � �8
�x � y � 9x � y � 1�12r � 6t � 0
r � t � 7
��3a � b � 79a � 8b � 4�5x � 3y � 8
x � 5y � �2
Reteaching
4.4 Solving Systems With Matrix Equations
Solve each system by writing and solving a matrix equation.Write the matrix equation that represents each system, andsolve the system, if possible, by using a matrix equation.
For each parabola, tell a. whether it opens up or down; b. whether it has a maximum or minimum value; c. where itcrosses the x-axis; and d. the coordinates of the vertex.
NAME _________________________________________________ CLASS _______________ DATE ______________
54 Reteaching 5.1 Algebra 2
◆Skill B Graphing a quadratic function and finding its maximum or minimum value
Recall In the form , if , the parabola opens up; if , theparabola opens down.
◆ ExampleDetermine for the graph of a. whether it opens up or down.b. whether it has a maximum or minimum value.c. where it crosses the x-axis.d. the coordinates of the vertex.
◆ Solutiona.
Since and , the parabola opens up.b. A parabola that opens up has a minimum value.c. If , then .
If , then .When 0, the graph crosses the x-axis. So this graph crosses the x-axis when and when .
d. The x-coordinate of the vertex is halfway between 2 and 4, so the axis of symmetry is the line .Substitute 1 for x in .
The coordinates of the vertex are (1, �9).Check with a graphics calculator.
� �9f(1) � (1 � 2)(1 � 4)
(x � 2)(x � 4)f(x) �x � 1
�
x � �2x � 4
f(x) �(x � 2)(x � 4) � (�2 � 2)(�2 � 4) � 0x � �2
(x � 2)(x � 4) � (6)(0) � 0x � 4
1 � 0a � 1f(x) � (x � 2)(x � 4) � x2 � 2x � 8
(x � 2)(x � 4)f(x) �
a � 0a � 0ax2 � bx � cf(x) �
x
y
O
2
6
10
–10
2 6–6 10–10
Lesson 4.3
1. inverses 2. not inverses
3.
4.
5.
6.
7.
8. The inverse does not exist.
9.
10.
11.
Lesson 4.4
1.
2.
3.
4.
5.
6.
7. (1, 1) 8. (4, �5) 9.
10. (3, 3, �4)
Lesson 4.5
1.
2.
3. (2, 1)
4.
5. (2, 4)
6. dependent system
7. (1.1, 0.3) 8. (0, 2, �3)
Reteaching — Chapter 5
Lesson 5.1
1.
2.
3.
4.
5.
6.
7. ; a � 1, b � �5, c � �24
8. ; a � 1, b � �16, c � 63
9. ; a � 4, b � 0, c � �49
10. ; a � 12, b � 1, c � �1
11. ; a � 49, b � 42, c � 9
12. ; a � 9, b � 0, c � �121
13. a. opens upb. minimumc. x-intercepts: �1 and 5d. (2, �9)
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 5.2 55
◆Skill A Solving a quadratic equation by taking square roots
Recall If , then and 4 or 4. The solutions are �4 and 4.
◆ Example 1Solve . Give exact solutions and approximate solutions to thenearest hundredth.
◆ Solution
Add 12 to each side.Divide each side by 5.Take the square root of each side.Use a calculator.
You can check on a graphics calculator by graphing and 188 onthe same screen. The x-coordinates of the intersections of these two graphs willbe the two solutions for the original equation.
◆ Example 2
Solve . Give exact solutions and approximate solutions to the
nearest hundredth.
◆ Solution
Multiply each side by 2.
Take the square root of each side.
or exact solutions
or approximate solutionsx � �0.16x � 6.16
3 � �10x � 3 � �10
x � 3 �10
x � 3 � �10
�(x � 3)2 � 10
(x � 3)2 � 10
12
(x � 3)2 � 5
12
(x � 3)2 � 5
y �y � 5x2 � 12
� 6.32x � �40
x2 � 405x2 � 200
5x2 � 12 � 188
5x2 � 12 � 188
x � �x �x � �16x2 � 16
Solve each equation. Give both exact solutions and approximateirrational solutions to the nearest hundredth.
1. 2. 3.
4. 5. 6. 5(x � 1)2 � 8012
(x � 5)2 � 3013
(x � 4)2 � 12
3x2 � 8 � 225x2 � 65x2 � 60
Reteaching
5.2 Introduction to Solving Quadratic Equations
Find the unknown length in each right triangle. Round answersto the nearest hundredth.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 5.3 57
◆Skill A Factoring a quadratic expression
Recall To factor a quadratic expression, start by looking for the following patterns.
(1) common monomial factor(2) difference of two squares(3) perfect square trinomial
◆ ExampleFactor each expression.a. b. c.
◆ Solutiona. common monomial factor
difference of 2 squaresb. common monomial factor
perfect square trinomialc. None of the basic 3 patterns occurs.
Find the factors of 24. → 1 24, 2 12, 3 8, 4 6Find a pair such that one factor is negative, so that the product is �24,and the sum of the factors is 5.
NAME _________________________________________________ CLASS _______________ DATE ______________
60 Reteaching 5.4 Algebra 2
◆Skill B Writing a quadratic function in the vertex form
Recall The equation can be written in the form , where(h, k) is the vertex of the parabola.
◆ ExampleGiven ,a. write the equation in vertex form.b. identify the transformations of the graph of .c. give the coordinates of the vertex.d. write the equation of the axis of symmetry.
◆ Solutiona.
Complete the square for by adding 4. To keep the equation in balance you must also add �4.
Notice that the standard vertex form contains �h. Therefore, in this problem , and .
b. This is a translation of by 2 units to the right and 3 units down.
c. The vertex of the parabola is at (h, k) � (2, �3).d. The axis of symmetry is the vertical line containing the vertex, .x � 2
x2f(x) �
k � �3h � 2a � 1
(x � 2)2 � 3g(x) �
(x2 � 4x � 4) � 1 � 4g(x) �
x2 � 4x
x2 � 4x � 1g(x) �
x2f(x) �
x2 � 4x � 1g(x) �
y � a(x � h)2 � ky � ax2 � bx � c
x
y
O
2
2–6–10
–6
–10
6 10
Lesson 5.4
1. 9 2. 64 3.
4. ;
5. ; 1.27 and 4.73
6. �5 and 3
7. ; 1.54 and 8.46
8. ; 0.80 and 11.20
9. �11 and 1
10. vertex form: vertex: (0, 0)axis of symmetry:
11. vertex form: vertex: (0, 5)axis of symmetry:
12. vertex form: vertex: (�2, �4)axis of symmetry:
13. vertex form: vertex: (1, �4)axis of symmetry:
14. vertex form: vertex: (�3, 0)axis of symmetry:
15. vertex form:
vertex:
axis of symmetry:
Lesson 5.5
1. 2.24 and �6.24
2. 1 and �3.5
3. 1.85 and �4.85
4. 0.68 and �0.88
5. 2.04 and �1.75
6. 0.75 and �0.5
7. 1.25 and �1
8. 0.59 and 5.08
9. x-intercepts: �3 and 5axis of symmetry: vertex: (1, �16)
10. x-intercepts: �5 and �1axis of symmetry: vertex: (�3, �4)
11. x-intercepts: 0 and 3axis of symmetry: vertex: (1.5, 2.25)
12. x-intercepts: 0.5 and �2.5axis of symmetry: vertex: (�1, �9)
13. x-intercepts: �2 and 3axis of symmetry: vertex: (0.5, 6.25)
14. x-intercept: 1.5axis of symmetry: vertex: (1.5, 0)
a � b � c � 0a(1)2 � b(1) � c � 0f(x) �x � 1a � b � c � 14a(�1)2 � b(�1) � c � 14f(x) �x � �1,(�1, 14)
ax2 � bx � c � f(x)
Use a graphics calculator to find the best quadratic model torepresent each set of data. If necessary, round coefficients to thenearest hundredth. Then graph the data and the regressionequation.
NAME _________________________________________________ CLASS _______________ DATE ______________
66 Reteaching 5.7 Algebra 2
◆Skill B Finding the quadratic model that best represents a set of data(You will need a graphics calculator.)
Recall A quadratic model will be in the form .
◆ ExampleLynda dives off the high diving board. You and Carlo use a stopwatch andobserve markings on the side of the board to calculate Lynda’s height above thewater at various times. As Carlo clocks times of 0.5 seconds, 1 second, 1.5seconds, and 2 seconds, you estimate Lynda’s altitude as 24 meters, 25 meters, 22 meters, and 17 meters respectively. Find the quadratic regression equation that best fits these data.
◆ SolutionUse a graphics calculator and enter the data points (0.5, 24), (1, 25), (1.5, 22), and (2, 17).Then use the quadratic regression feature to find a quadratic model.
Graph the data points and the regression equation. Notice that the data do not exactly fit the curve, but your estimates of height were actually quite accurate.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 5.8 67
◆Skill A Graphing a quadratic inequality in one variable
Recall The product of two factors is positive only if both factors have the same sign.
◆ Example 1Graph the solution to on a number line.
◆ Solution
Since this product is 0 if or , mark these points on the number line.The product is positive if both factors are positive or both factors are negative. Sofind the values of x that cause both factors to have the same sign.
only when .
only when .
only whenor .
Therefore the product is greater than or equal to 0, that is only if
or . The graph is shown below.
◆ Example 2Graph the solution to on a number line.
◆ SolutionUse the chart in Example 1 to see that the product is negative, that is
, only if .
0 2 4 6 8 10–2–4–6–8–10x
�6 � x � 2(x � 6)(x � 2) � 0
x2 � 4x � 12 � 0
0 2 4 6 8 10–2–4–6–8–10x
x � 2x � �6(x � 6)(x � 2) � 0
x � 2x � 6(x � 6)(x � 2) � 0
x � 2x � 2 � 0
x � �6x � 6 � 0
x � 2x � �6(x � 6)(x � 2) � 0x2 � 4x � 12 � 0
x2 � 4x � 12 � 0
Graph the solution to each inequality on a number line.
NAME _________________________________________________ CLASS _______________ DATE ______________
68 Reteaching 5.8 Algebra 2
◆Skill B Graphing a quadratic inequality in two variables
Recall If a quadratic function is written in the form , the vertex is atthe point (h, k).
◆ Example 1Graph .
◆ SolutionStart with the equation .
Complete the square.vertex form
The vertex is at (�1, �3).
For shade above the curve.The shaded region is where the y-coordinate of each point is greater than for the x-coordinate of that point.
◆ Example 2Graph .
◆ SolutionDraw the parabola from Example 1 with a broken curve and shade below the curve.Test the point (0, 0).Since , this point is not in the shaded region.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 6.1 69
◆Skill A Finding the multiplier for growth or decay
Recall A multiplier greater than 1 models growth. A multiplier between 0 and 1 modelsdecay.
◆ ExampleFind the multiplier for each situation.a. 5% growth b. 8% decay
◆ Solutiona. Add the growth rate to 100%.
100% � 5% � 105% or 1.05The multiplier is 1.05.
b. Subtract the rate of decay from 100%.100% � 8% � 92% or 0.92
The multiplier is 0.92.
Find the multiplier for each situation.
1. 12% growth 2. 25% decay 3. 7.5% decay
4. 8.2% growth 5. 1% growth 6. 0.5% decay
Reteaching
6.1 Exponential Growth and Decay
◆Skill B Writing and evaluating an exponential expression that models growth or decay(You will need a calculator.)
Recall Any growth or decay rate related to a natural event assumes that the rate remainsconstant, and a prediction based on this rate will give approximate results.
◆ Example The population of a small town of 10,000 people is growing at the rate of about5.2% per year. Predict the approximate population 10 years from now.
◆ SolutionThe multiplier is 100% � 5.2% � 105.2% or 1.052.
10,000(1.052)10 ≈ 16,602.The predicted population is about 16,600.
Use a growth or decay model to solve each problem.A new school district is experiencing an annual growth rate of 9.5%. Theschool population is now 5600 students. What is the approximate predictedpopulation
7. 3 years from now? 8. 5 years from now? 9. 10 years from now?
Use a calculator and table to solve each problem.
13. After 2 hours, only 75% of a new medication remains in your body. If you take an 80-milligram tablet, and this rate of decay is constant, in approximately how many hours will less than 15 milligrams remain in your system?
14. You invest $5000 in an account that earns interest at an effective rate of 8.4% per year. In how many years will you have over $6800 in the account?
15. If you invest $50,000 in a high interest account that earns interest at an effective rate of 13.8% per year, how many years will it take to double your money?
NAME _________________________________________________ CLASS _______________ DATE ______________
70 Reteaching 6.1 Algebra 2
◆Skill C Using a table to find a specific value for an exponential function(You will need a calculator.)
◆ Example Your doctor prescribes a medication for your allergies. After each 1 hour interval,only 90% of the medication present 1 hour ago remains in your system. If youtake a 100-milligram tablet, in approximately how many hours will only 50% ofthe medication remain in your system?
◆ SolutionThe multiplier is 100% � 10% � 90%, or 0.9.Make a table for 100(0.9)x, where x is a positive integer. Use a calculator.
50% of the medication will be left in your system between 6 and 7 hours after theinitial dose.
x 1 2 3 4 5 6 7
100(0.9)x 90 81 72.9 65.61 59.05 53.14 47.83
The rate in the number of reported cases of robbery is dropping at about 7%per year in a given region of the country. The number of cases reported thisyear was approximately 156,000. If the number continues to drop at this rate,what is the approximate predicted number of cases
10. 1 year from now? 11. 3 years from now? 12. 5 years from now?
NAME _________________________________________________ CLASS _______________ DATE ______________
80 Reteaching 6.6 Algebra 2
◆Skill B Finding interest compounded continuously
Recall If the principal, P, is invested at an annual interest rate of r, compounded continuously, the amount, A, in the investment after t years is A � Pert.
◆ Example 1Find the amount in an account if $1500 is invested at an annual rate of 5.8% andinterest is compounded continuously for 7 years.
◆ SolutionA � Pert
A � 1500e(0.058)(7) 5.8% � 0.058A � $2251.20 Use a calculator.
◆ Example 2How long will it take to double your money if you deposit $500 at an annual rateof 7.2% compounded continuously?
◆ SolutionA � Pert
1000 � 500e(0.072)t $500 doubles to $10002 � e(0.072)t Divide each side by 500.
ln 2 � ln e(0.072)t Take the natural logarithm of each side.ln 2 � 0.072t inverse functions: ln ex � x
t ≈ 9.63It will take about 9 years and 7.5 months to double your money.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 6.7 81
◆Skill A Solving problems using an exponential growth function
Recall The continuous exponential growth function is P(t) � P0ekt, where population
depends on time and P0 is the initial population.
◆ Example 1A population of bacteria grows exponentially. An initial population of 5000bacteria grows to 18,000 in 3 hours. Write an exponential function for thispopulation of bacteria in terms of time.
◆ SolutionP(t) � P0e
kt where P0 � 5000Since the population is 18,000 after 3 hours, t � 3 and P(3) � 18,000.18,000 � 5000ek(3)
Divide each side by 5000 and simplify.
k ≈ 0.427Therefore, P(t) ≈ 5000e0.427t.
◆ Example 2If the bacteria continue to grow at this rate, predict the approximate populationof bacteria in above example 24 hours after the initial time.
◆ SolutionP(t) ≈ 5000e0.427t
P(24) ≈ 5000e0.427(24) t � 24≈ 141,130,167 or approximately 140,000,000.
k �ln( 18
5 )3
ln(185 ) � 3k
ln(185 ) � ln e3k
185
� e3k
Use the exponential growth function, P(t) = P0ekt, to complete
the following.
1. A population of bacteria grows exponentially. An initial population of 1000 bacteria grows to 2500 in 4 hours. Write the function for this population of bacteria as a function of time. Then estimate the population at the end of 16 hours.
2. A special savings account grew exponentially due to interest from$5000 to $5682 in one year. If the account continues to grow at this rate, how much will be in the account 5 years later?
Reteaching
6.7 Solving Equations and Modeling
Solve each equation algebraically and graphically.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 7.2 85
◆Skill A Describing the graph of a polynomial function(You will need a graphics calculator.)
Recall Graphs are always read from left to right.A function is increasing from left to right if the y-values are increasing.A function is decreasing from left to right if the y-values are decreasing.
◆ Example Given :a. graph the function,b. give the approximate coordinates of each local minimum and maximum,c. find the intervals over which the function is increasing, andd. find the intervals over which the function is decreasing.
◆ Solutiona. The graph is shown at right.b. Local minimum: approximately
(�0.8, �6.2)For all values of x close to �0.8, the y-coordinate is greater than �6.2.Local maximum: approximately (2.1, 6.1)For all values of x close to 2.1, the y-coordinate is less than 6.1.
c. increasing when d. decreasing when and also when x � 2.1x � �0.8
�0.8 � x � 2.1
P(x) � �x3 � 2x2 � 5x � 4
Graph each function. Find any local maxima or minima to the nearest tenth, and the intervals over which the function is increasing and decreasing.
NAME _________________________________________________ CLASS _______________ DATE ______________
86 Reteaching 7.2 Algebra 2
◆Skill B Describing the end behavior and the number of turning points of the graph of apolynomial function (You will need a graphics calculator.)
Recall The degree of a polynomial is determined by the greatest power of the variable.
◆ Example Describe the end behavior and state the number of turning points for each graph shown below.
◆ Solutionf(x): As you read across the graph from
left to right, this function is rising on the left and on the right. There are 2 turning points in between.
Cubic functions have at most 2 turning points.
g(x): As you read across the graph from left to right, this function is rising on the left but falling on the right with one turning point.
Quadratic functions have one turning point.
–10 10
10
–10
f(x)
g(x)
10. Exercise 4 is an odd-degree function(degree 3) and the coefficient of x3 ispositive (�1). Using this as a pattern,predict the end behavior and greatestpossible number of turning points of thegraph of the function
.
12. Using the even-degree function in Exercise6 as a model, predict the end behavior andgreatest possible number of turning pointsof the graph of the function
.
11. Exercise 5 is an odd-degree function andthe coefficient of x3 is negative. Using thisas a pattern, predict the end behavior andthe greatest possible number of turningpoints of the graph of the function
.
13. Using the even-degree function in Exercise7 as a model, predict the end behavior andgreatest possible number of turning pointsof the graph of the function
.f(x) � �x4 � 3x
f(x) � �x5 � 2x2 � 1
f(x) � x4 � 4x3 � 2x2 � 12x � 1
f(x) � x5 � 15x4 � 85x3 � 225x2 � 274x � 120
16.
quadratic function; 1 turning point atapproximately 2
17.
cubic function; 2 turning points atapproximately 0 and 2.5
Lesson 7.2
1.
maximum at (�2.7, 2.5); minimum at (0, �7); x � �2.7, x > 0;�2.7 � x � 0
2.
maximum at (0, 5); minimum at (�1.2, 2.8) and (1.2, 2.8);�1.2 � x � 0 and x � 1.2; x � �1.2 and 0 � x � 1.2
3.
maximum at (1.8, 8.1); no minimum; x � 1.8; x � 1.8
4. rising on the left and the right,2 turning points
5. falling on the left and the right,2 turning points
6. falling on the left, rising on the right,1 turning point
7. rising on the left, falling on the right,1 turning point
8. rising on the left and the right,4 turning points
9. rising on the left, falling on the right,1 turning point
10. rising on the left and the right,4 turning points
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 7.4 89
◆Skill A Using graphs, synthetic division, and factoring to find rational roots
Recall If 3 is a zero of , then is a solution (root) ofand is a factor of .
◆ Example Find the roots of .
◆ SolutionGraph the function and check for zeros of the function. (These occur where the graph crosses the x-axis.) The graph indicates that 2 may be a root of the given equation.
Use synthetic division to check if 2 is a root.
2 1 0 �3 �22 4 2
1 2 1 0 ← The zero remainderindicates that 2 is a root.
This means that is a factor of .
Thus, . (You read the coefficients of thesecond factor, , from the first three numbers, 1, 2, and 1, found in thelast line of the synthetic division above.)
Factor
Solve .
Zero-Product Property
The roots are 2 and �1, with �1 occurring twice.
x � �1x � �1x � 2x � 2 � 0 or x � 1 � 0 or x � 1 � 0(x � 2)(x � 1)2 � 0(x � 2)(x � 1)2 � 0
x2 � 2x � 1: x2 � 2x � 1 � (x � 1)2
x2 � 2x � 1x3 � 3x � 2 � (x � 2)(x2 � 2x � 1)
x3 � 3x � 2x � 2
f(x) � x3 � 3x � 2
x3 � 3x � 2 � 0
x2 � x � 12x � 3x2 � x � 12 � 0x � 3P(x) � x2 � x � 12
Use a graph, synthetic division, and factoring to find all of theroots of each equation.
1. 2.
3. 4.
5. 6. x3 � 16x � 0x3 � 4x2 � 28x � 32 � 0
x3 � 15x2 � 75x � 125 � 02x3 � x2 � 54x � 72 � 0
x3 � 7x � 6 � 04x3 � 8x2 � 11x � 3 � 0
Reteaching
7.4 Solving Polynomial Equations
–10 10
10
–10
Use variable substitution to find all of the roots of each equation.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 7.5 91
◆Skill A Finding all the zeros of a polynomial function
Recall A corollary to the Fundamental Theorem of Algebra states that an nth-degreepolynomial function will have exactly n complex zeros.
◆ Example Find all the zeros of .
◆ SolutionThe only possible rational roots are those found by using a factor of −9 (constantterm) as the numerator of a fraction and a factor of 2 (coefficient of x4) as thedenominator.
possibilities:
The graph shows that one zero appears tobe �1.
�1 2 �1 3 �3 �9�2 3 �6 9
2 �3 6 �9 0
The zero remainder indicates that �1 is a zero.The graph shows that there is another zero between 2 and 3. The only possible rational root from the list above between
2 and 3 is , or 1.5.
To see if 1.5 is a root, apply synthetic division to found fromthe last line of the synthetic division above.
1.5 2 �3 6 �93 0 9
2 0 6 0 ← zero remainder
So 1.5 is also a zero.
This leaves a last row (without the remainder) which represents .
Use the quadratic formula where a � 1, b � 0, and c � 6.
The four zeros of are −1, 1.5, , and .�i�3i�3f(x) � 2x4 � x3 � 3x2 � 3x � 9
x ��0 �02 � 4(2)(6)
2(2)�
��484
� 4i�3
4� i�3
2x2 � 6
2x3 � 3x2 � 6x � 9
32
11
, 12
, 31
, 32
, 91
, 92
f(x) � 2x4 � x3 � 3x2 � 3x � 9
Find all of the rational roots of each polynomial equation.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 8.1 93
◆Skill A Identifying and using direct and inverse variations
Recall Direct variation can be modeled by ; inverse variation can be modeled by
.
◆ Example 1Determine which type of variation is demonstrated by each of the following.a. the distance you travel when riding at a constant speedb. the time it takes to get to your destination if you accelerate
◆ Solutiona. Your distance increases as your time increases. This is an example of direct
variation.b. The time it takes to get to your destination decreases as your speed increases.
This is an example of inverse variation.
◆ Example 2If y varies inversely as x, and 15 when 4, find the value of y when 10.
◆ Solution
or
, so .
When 6.�x � 10, y �6010
y �60x
k � 4 � 15 � 60
k � xyy �kx
x �x �y �
y �kx
y � kx
Determine whether each of the following is an example of directvariation or inverse variation.
1. the number of words you can type in 30 minutes as typing speed increases
2. the number of tickets you can buy for $100 as the price per ticket increases
3. the number of credit-hours you have as you take more courses
For each problem, assume that y varies inversely as x.
4. If 10 when 10, 5. If 1 when 12, 6. If y � 7 when x � 14,find y when 4. find y when 48. find y when 7.
7. If 2.5 when 1.5, 8. If 100 when 0.4 9. If when ,find y when 4. find y when 10,000.
find y when .x �13
x �x �x �
14
y �23
x �y �x �y �
x �x �x �x �y �x �y �
Reteaching
8.1 Inverse, Joint, and Combined Variation
Solve each variation problem.
10. If y varies directly as x2, and 20 11. If y varies inversely as , and 10 when 2, find y when 7. when 25, find y when 100.
12. If y varies jointly as x and z, and 49 13. If y varies directly as x and inversely as ,when 2.8 and 3.5, find y when and 4 when 12 and 3, find y
1.6 and 7.5. when 7 and 5.
14. Newton’s Universal Law of Gravity is given as . If G is a
constant, explain how F varies in relation to the variables m1, m2, and r.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 8.2 95
◆Skill A Finding the domain of a rational function
Recall Division by zero is not allowed; it is undefined.
◆ Example
Find the domain of .
◆ SolutionYou must exclude from the domain any values of x which cause the denominatorto have a value of 0.Set equal to 0.
orThe domain is all real numbers except �5 and 2.
x � 2x � �5(x � 5)(x � 2) � 0
x2 � 3x � 10
g(x) �x2 � 8x
x2 � 3x � 10
Find the domain of each rational function.
1. 2. 3. x2 � 5x2 � 4x � 21
5xx2 � 7x
x � 3x2 � 16
Reteaching
8.2 Rational Functions and Their Graphs
◆Skill B Identifying vertical asymptotes and holes in the graph of a rational function
Recall If is a factor in both the numerator and denominator, there will be a hole inthe graph at . If is a factor of the denominator but not a factor of thenumerator, there will be a vertical asymptote of .
◆ Example
For the rational function
a. identify the x-coordinates of any holes in the graph.b. write the equations of any vertical asymptotes.
◆ Solution
a.
Since is a factor of both the numerator and denominator, there will be a hole at .
b. Since is a factor of the denominator but not the numerator, and has a value of 0 when 3, there will be a vertical asymptote at
3.x �
x �
x � 3x � �2
x � 2
2x2 � 3x � 2x2 � x � 6
�(2x � 1)(x � 2)(x � 2)(x � 3)
f(x) �2x2 � 3x � 2x2 � x � 6
x � b(x � b)x � b
(x � b)
–10 10
10
–10
Identify all holes and asymptotes in the graph of each rationalfunction.
NAME _________________________________________________ CLASS _______________ DATE ______________
96 Reteaching 8.2 Algebra 2
◆Skill C Writing an equation for the horizontal asymptote of a graph
Recall The degree of a polynomial is the greatest degree of its terms.
◆ ExampleWrite the equation of the horizontal asymptote for each of the followingfunctions.
a. b. c.
◆ Solutiona. Since the degree of the numerator and denominator are the same, divide the
coefficient of the term with the greatest degree in the numerator by thecoefficient of the like term in the denominator.
(coefficients of the terms)
Thus, 2 is a horizontal asymptote. The graph is shown on the preceding page.
b. Since the degree of the numerator is less than the degree of the denominator,the horizontal asymptote is 0. (The graph is shown at left below.)
c. Since the degree of the numerator is greater than the degree of thedenominator, there is no horizontal asymptote. (The graph is shown at right below.)
y �
y �
x221
� 2
h(x) ��x2
x � 2g(x) �
3xx2 � 5
f(x) �2x2 � 3x � 2
x2 � x � 6
Identify any horizontal asymptotes for the following functions.Use a graphics calculator to check your answer.
8. 9. 10. x � 5(x � 1)(x � 4)
f(x) �x2 � 5x � 6
x � 2f(x) �
5x2 � 82x2 � 3x
f(x) �
–10 10
10
–10
–10 10
10
–10
Lesson 8.2
1. all real numbers,
2. all real numbers,
3. all real numbers,
4. hole at x � �2; vertical asymptote: x � �3
5. no holes; vertical asymptotes: x � �4 and x � 1
NAME _________________________________________________ CLASS _______________ DATE ______________
100 Reteaching 8.4 Algebra 2
◆Skill C Adding and subtracting rational expressions with unlike denominators
Recall To add or subtract two fractions, they must have a common denominator.
◆ Example
Simplify.
◆ SolutionFactor the denominators: and .Then find the LCM of and . The LCM is .Multiply each fraction by 1 so that they will both have the same denominator.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 8.5 101
◆Skill A Solving rational equations
Recall You should find the domain before solving the equation.
◆ Example
Solve.
◆ Solution1. Factor so that you can determine the domain.
The domain is all real numbers except 3 and �2.2. The common denominator for all of the fractions is .3. Multiply every term by the common denominator.
4. Simplify this equation so that no fractions remain.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 8.6 103
◆Skill A Evaluating cube-root expressions
Recall and
◆ ExampleEvaluate each expression.
a. b. c.
◆ Solution
a. b. c.
� �15� 3� 5(�3)� 23
�13
� 3213
3�272� 5 3��275( 3��1084 )� 42 � 7( 3�64)2 � 7
13
3�2725( 3��1084 )( 3�64)2 � 7
x23 �
3�x2 � ( 3�x)2x13 �
3�x
Evaluate each expression without using a calculator.
1. 2. 3.
4. 5. 6. (274)13�1
2 3��216��11
3( 3�813 )
(�8)13 � 2 3�1252 ( 3�1252)
Reteaching
8.6 Radical Expressions and Radical Functions
◆Skill B Using transformations to graph square root functions
Recall The domain of a square root function is the set of all x such that the radicand(expression under the radical sign) is greater than or equal to 0.
◆ ExampleFor the function a. state the domain andb. use transformations to sketch a graph.
◆ Solutiona. Since the radicand must be
nonnegative, . The domain is .
b. Write in the form , where
is the vertical stretch/compression factor,is the horizontal stretch/compression factor,
h gives the horizontal translation, and k gives the vertical translation.In this function, , and .
Start with and stretch vertically by a factor of 3; reflect across the x-axis(since ); there is not a horizontal stretch; translate 1 unit to the left and 5units up.
a � 0�x
�3�1(x � (�1)) � 5g(x) �
k � 5a � �3, b � 1, h � �1
�b��a�
a�b(x � h) � kg(x) �
�3�x � 1 � 5g(x) �x � �1
x � 1 � 0
�3�x � 1 � 5g(x) �
–10 10
10
–10
g(x)
f(x)
Find the inverse of each quadratic function. Then graph thefunction and its inverse in the same coordinate plane.
Find the center, the circumference, and the area of each circledescribed below.
7. The endpoints of a diameter are (2, �1) and 8. The endpoints of a diameter are (0, �6) (�6, 5). and (5, 6).
a. a.
b. b.
c. c.
9. The endpoints of a diameter are (0, 0) and 10. The endpoints of a diameter are (�3, �10)(8, 8). and (6, 2).
a. a.
b. b.
c. c.
Find the distance between P and Q and the coordinates of themidpoint of . Give exact answers and approximate answers tothe nearest hundredth when appropriate.
4. P(7, 1) and Q(6, 2) 5. P(�2, �5) and Q(�2, 4) 6. P(�3, 6) and Q(1, �2)
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 9.2 111
◆Skill A Graphing a parabola and labeling its parts
Recall If the vertex of a parabola is at (h, k) and the distance between the vertex and thefocus is p, then the following are true.
◆ ExampleGraph . Label the vertex, focus, and directrix.
◆ Solution
Complete the square for the y-terms.
Divide by 12.
In this form you can see that the vertex is at
(�3, 2) and, since , the value of p is 3.
The graph opens to the right.Graph the vertex at (�3, 2) and mark the point 3 units to the right, (0, 2), as the focus.The directrix is also at a distance p from the vertex, so its equation is .
Make a table of x- and y-coordinates to find several other points.
x � �6
14p
�1
12
x � 3 �1
12(y � 2)2
12x � 36 � (y � 2)212x � 32 � y2 � 4y
y2 � 12x � 4y � 32 � 0
y2 � 12x � 4y � 32 � 0
Graph each equation. Label the vertex, focus, and directrix. Check with a graphics calculator.
1. 2. 3.
x
y
Ox
y
Ox
y
O
x2 � 8x � y � 18 � 0x � �18
(y � 1)2 � 2y �14
(x � 2)2
Reteaching
9.2 Parabolas
p � 0 opens up opens to the right
p � 0 opens down opens to the left
x � h �14p
(y � k)2y � k �14p
(x � h)2
x
y
O
2
–2
4
directrixx = –6
FV
2 4–4
Write the standard equation for each parabola below.
NAME _________________________________________________ CLASS _______________ DATE ______________
116 Reteaching 9.4 Algebra 2
◆ Solution
Complete the squares.
Standard Form
(Recall that a2 � b2.)Therefore, the center is at (2, �1).Since is in the x-term, the major axis is horizontal.The foci are at a horizontal distance c from the center:
and .
The vertices are at a horizontal distance a from the center: and (5, �1).The co-vertices are at a vertical distance b from the center: (2, 0) and (2, �2).
Recall To sketch a graph of any conic, write its equation in standard form.
◆ Example
Sketch a graph of .
◆ SolutionThe center is at (�2, 0). Since 9 is in the x-term, mark points at a distance horizontally from the center; (�5, 0) and (1, 0) are the vertices.Since 25 is in the y-term, mark points at a distance vertically from the center; (�2, 5) and (�2, �5) are the co-vertices.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 9.5 117
◆Skill A Writing an equation for a graphed hyperbola
Recall Hyperbola with a horizontal transverse axis:
Hyperbola with a vertical transverse axis:
◆ ExampleFor the graph at right, write an equation fora. the hyperbola. b. each asymptote.
◆ Solutiona. The center is at (h, k) (0, 2).
Since the transverse axis is horizontal,
.
This implies that a is the horizontal distance from the center to the vertices, so 2.This leaves 3 as the vertical distance from the center to the edge of the “rectangle.” The equation of the hyperbola is
.
b. The asymptotes both have a y-intercept of
2 and slopes of and .
The equations of the asymptotes are and .y � �32
x � 2y �32
x � 2
�ba
� �32
ba
�32
x2
4�
(y � 2)2
9� 1
b �a �
(x � 0)2
a2 �(y � 2)2
b2 � 1
�
(y � k)2
a2 �(x � h)2
b2 � 1
(x � h)2
a2 �(y � k)2
b2 � 1
Write the standard equation for each hyperbola and give theequations for the asymptotes.
NAME _________________________________________________ CLASS _______________ DATE ______________
120 Reteaching 9.6 Algebra 2
◆Skill B Solving a nonlinear system by graphing
Recall A coordinate graph will show only the real-number solutions for a system.
◆ Example
Solve by graphing.
◆ SolutionSolve the first equation for y.
Use a graphics calculator to graph , and .Use the trace or intersection feature of your calculator to find solutions to the nearest tenth.The solutions are (2.9, 2.7), (�2.9, 2.7), (1.5, �3.7), and (�1.5, �3.7).
y3 � x2 � 6y1 � �16 � x2, y2 � ��16 � x2
y � �16 � x2
y2 � 16 � x2
�x2 � y2 � 16y � x2 � 6
–10 10
10
–10
Classify the conic section defined by each equation.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.1 121
◆Skill A Applying the Fundamental Counting Principle
Recall If there are m ways to make one selection and n ways to make another selection,there are ways to make both selections.
◆ Example 1In your daily commute to school you find that there are 5 different routes to theexpressway and 3 different routes from the expressway to school. How manydifferent trips could you take to get to school?
◆ Solution5 routes to expressway3 routes from expressway to school
There are different routes to school.
◆ Example 2Tim’s security password on a computer at school must consist of 4 letters. How many different passwords can he choose if he doesn’t use any letter more than once?
◆ Solutionchoices for 1st letter: 26choices for 2nd letter: 25choices for 3rd letter: 24choices for 4th letter: 23
There are or 358,800 different possible codes.26 � 25 � 24 � 23
5 � 3 � 15
m � n
Use the Fundamental Counting Principle to answer the following.
1. A pizza outlet offers 6 kinds of meat toppings and 10 different vegetabletoppings. How many different pizzas with 1 meat and 1 vegetable could you order?
2. In a word game, you may replace 2 of the letters on your tray on yourturn. The letters you have are E, X, J, U, F, A, and Z. In how many ways could you select 1 consonant and 1 vowel to discard?
3. A local department store will put a 3-letter monogram on your selection of towels. How many different monograms are possible?
4. You must decide how to appoint 9 people to the 9 different positions on a baseball team. In how many different ways could you do this?
5. How many different signals can be shown by arranging 3 flags in a row if 7 different flags are available?
Reteaching
10.1 Introduction to Probability
Find the probability of each event.
6. If you flip a quarter, what is the probability it will land a. heads up? b. tails up?
7. If you roll a number cube, what is the probability of rolling
a. 6? b. a number less than 6? c. 7? d. a number less than 7?
8. A 3-digit number is chosen at random. Assuming the first digit is not 0,what is the probability that the number is
a. a multiple of 5? b. a multiple of 5 and less than 400?
9. A multiple-choice test has 50 questions. Each question has 5 choices. For each question, you make a choice at random.
a. What is the probability that you will get the first answer correct?
b. Theoretically, how many answers might you expect to get correct?
NAME _________________________________________________ CLASS _______________ DATE ______________
122 Reteaching 10.1 Algebra 2
◆Skill B Finding the theoretical or experimental probability of an event
Recall The probability of an event is always between 0 and 1 inclusive; .
◆ Example 1A drawer contains 6 brown, 8 blue, and 4 tan single socks. If you randomly pullout 2 socks, a. what is the probability that the first sock is blue?b. if the first sock is blue, what is the probability that the second one is blue?
◆ Solutiona. Since there are a total of 18 socks, the theoretical probability of getting a blue
one is , or approximately 44%.
b. Since there are only 17 socks left and 7 of these are blue,
, or approximately 41%.
◆ Example 2A testing center has randomly assigned you a 4-digit number. Assuming the firstdigit is not 0, what is the probability that this number is even and is greater thanor equal to 3000?
◆ SolutionStart by using the Fundamental Counting Principle. Consider each digitseparately.
1st digit could be any digit from 3 to 9 7 choices2nd digit could be any of 10 digits 10 choices3rd digit could be any of 10 digits 10 choices4th digit must be 0, 2, 4, 6, or 8 5 choices
, or approximately 39%number of even numbers � 3000number of 4- digit numbers
�7 � 10 � 10 � 5
9 � 10 � 10 � 10�
718
number of blue sockstotal number of socks
�717
number of blue sockstotal number of socks
�8
18�
49
0 � P(event) � 1
Reteaching — Chapter 10
Lesson 10.1
1. 60 2. 12 3. 17,576
4. 362,880 5. 210
6. a. b.
7. a. b. c. 0 d. 1
8. a. b.
9. a. b. 10
Lesson 10.2
1. 40,320 2. 6,497,400
3. 60 4. 362,880
5. If this were a permutation of 26 letterstaken 3 at a time, you could not use anyletter more than once on a license plate.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.2 123
◆Skill A Solving problems involving linear permutations
Recall A permutation is an arrangement of objects in a specific order.
◆ Example 1In how many ways can you arrange 7 books on a shelf?
◆ SolutionAny one of 7 books can occupy the 1st position. For each of these, any one of the6 remaining books can be in the 2nd position. Therefore, for all 7 books there are
arrangements.
There are 5040 different ways to arrange the books.
◆ Example 2A CD-player is set to choose the tracks of a 15-track CD at random. In how manyways could 5 tracks be played?
◆ SolutionThere are 15 choices for the 1st track and 14 choices for the 2nd track. For all 5tracks there are choices.
Notice that , where
There are 360,360 ways to play 5 of the 15 tracks.
◆ Example 3Suppose a “word” can be composed of any set of letters in a particular order. Howmany different 5-letter “words” can you make from the letters RADAR using allof the letter in each “word?”
◆ Solution
But since the 2 R’s are indistinguishable from each other, and likewise for the 2A’s, you must “divide out” these arrangements.
You can make 30 different five-letter “words.”5!2!2!
1. In how many ways could you arrange 8 different shirts on hangers in the closet?
2. From a deck of 52 cards, how many different ways could you draw 4 cards,
if the order in which you draw them will make a difference?
Reteaching
10.2 Permutations
3. Find the number of permutations of all the letters in the word PEPPER.
4. How many different batting lineups are possible for 9 players on a baseball team?
5. Many license plates start with 3 letters of the alphabet. Consider all thepossible 3 letters that could be used. Why is this not equal to 26P3? (Hint: Consider the 3 letters AAA.)
6. The Hawaiian alphabet has only 12 letters. How many permutations could be made using 5 different letters?
7. Find the number of permutations of all the letters in the word MISSISSIPPI.
NAME _________________________________________________ CLASS _______________ DATE ______________
124 Reteaching 10.2 Algebra 2
◆Skill B Solving problems involving circular permutations
Recall A circular arrangement, unlike a linear arrangement, has no beginning or endingpoint.
◆ ExampleHow many different arrangements are there for seating 6 people around acircular table?
◆ SolutionSince a circular table has no “end,” one person could be anywhere around thetable. Relative to this person, the rest of the people could be arranged in 5!different ways.
There are 120 circular arrangements of the 6 people.
(n � 1)! � (6 � 1)! � 5! � 120
Find the number of permutations for each problem.
8. In how many orders can 8 children be standing on the playground merry-go-round?
9. A stage dance starts with 12 people in a circle. In how many different ways could these 12 dancers be arranged?
10. Five CDs are placed on a rotary tray in a CD-changer. In how many different ways could these CDs be arranged?
11. If 2 of the CDs in Exercise 10 were identical, in how many ways could all of the CD’s be arranged on the tray?
12. Two rings in a child’s playpen each have 4 large plastic beads strung onthem. One appears to have the beads in a red-blue-green-yellow orderand the other in a yellow-green-blue-red order. But you realize that ifyou flip the second ring over, it has exactly the same arrangement ofthese 4 colors as the other ring. Taking this fact into account, in how many different orders could these 4 beads be placed on the ring?
Reteaching — Chapter 10
Lesson 10.1
1. 60 2. 12 3. 17,576
4. 362,880 5. 210
6. a. b.
7. a. b. c. 0 d. 1
8. a. b.
9. a. b. 10
Lesson 10.2
1. 40,320 2. 6,497,400
3. 60 4. 362,880
5. If this were a permutation of 26 letterstaken 3 at a time, you could not use anyletter more than once on a license plate.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.3 125
◆Skill A Solving problems involving combinations and permutations
Recall For 3 distinct objects there are 3! permutations (arrangements) but only 1combination of all 3.
◆ ExampleYou have decided to read 5 novels out of a list of 12 for this semester’s English class.a. How many different sets of 5 novels could you choose?b. In how many different orders could you read the 5 novels you choose?
◆ Solutiona. There are 12P5 � 95,040 different arrangements of 5 books out of 12. But since
order is not being considered here, any set of 5 books could be chosen in 5! � 120 different orders. Therefore, to find the combinations of 5 books youmust “divide out” the 120 orders.
There are 792 combinations of 5 novels to read.
b. Once you have chosen 5 novels, there is only one combination of these 5, butthere are 5! � 120 different orders in which you could read them.
12P5
5!�
95,040120
� 792
Solve each combination problem.
1. You have 7 books, but only 3 will fit on the shelf.a. How many different arrangements of 3 books out of 7 could you make?
b. How many different ways could you select 3 of the 7 books, without regard to order?
c. How many ways could you arrange one selection of 3 books on the shelf?
2. A science class has 20 students.a. In how many ways could a group of 3 students be selected from the
class?
b. In how many ways could the whole class elect a reader, a recorder, and a grader for a class project?
3. Six people wish to play bridge. But only 4 of them can play at one time. Ifthey decide to play a game for each possible group of 4 players, how many games will they need to play?
4. You are dealt 4 cards for a card game. How many different 4-card hands are possible when starting a game with a 52-card deck?
5. The choices for a sandwich are 4 different meats and 5 different cheeses. How many different sandwiches of 2 meats and 2 cheeses could you make?
Reteaching
10.3 Combinations
Solve each combination problem.
6. From a deck of 52 cards, 4 cards are drawn. What is the probability that
a. all 4 cards are black? (A deck has 26 black cards.)
b. all 4 cards are spades? (A deck has 13 spades.)
c. all 4 cards are face cards? (A deck has 12 face cards.)
d. all 4 cards are aces? (A deck has 4 aces.)
7. Five apples, 7 oranges, and 4 peaches are mixed in a fruit bin. If 4 pieces offruit are picked out at random, what is the probability of picking
a. 2 oranges and 2 peaches? b. 4 oranges or 4 apples?
8. Out of 40 sketches submitted, 8 were picked at random. If you submitted5 sketches, what is the probability that exactly 2 of your sketches were picked?
9. From X, P, O, Z, E, A, D, F, M, and B, 3 letters are picked at random. What is the probability that
a. 3 consonants are picked? b. at least 1 vowel is picked?
NAME _________________________________________________ CLASS _______________ DATE ______________
126 Reteaching 10.3 Algebra 2
◆Skill B Using combinations and probability
Recall .
◆ ExampleFrom a small club of 8 girls and 6 boys, four students are to be chosen asrepresentatives to a city-wide convention. If the committee of 4 students ischosen at random with a hat drawing, what is the probability that it will havea. exactly 2 girls and 2 boys? b. at least 3 girls?
◆ Solutiona. There are 14C4 � 1001 combinations of 14 students taken 4 at a time.
There are 8C2 � 28 combinations of 2 girls and 6C2 � 15 combinations of 2 boys.
The probability of picking exactly 2 girls and 2 boys is approximately 42%
b. “At least 3 girls” means there could be 3 girls and 1 boy or 4 girls and no boys.Combinations of 3 girls and 1 boy � 8C3 � 6C1. (Note: “and” impliesmultiplication.)
Combinations of 4 girls and 0 boys � 8C4 � 6C0.
Number of combinations of 3 girls and 1 boy or 4 girls and 0 boys(8C3)(6C1) � (8C3)(6C0) (Note: “or” implies addition.)
Therefore, .�56 � 6 � 70 � 1
1001� 0.41, or 41%
(8C3)(6C1) � (8C3)(6C0)14C4
4201001
� 0.428C2 � 6C2
14C4�
(ways to choose 2 girls) � (ways to choose 2 boys)ways to choose 4 students
�
Probability �number of desired outcomes
total number of possible outcomes
Reteaching — Chapter 10
Lesson 10.1
1. 60 2. 12 3. 17,576
4. 362,880 5. 210
6. a. b.
7. a. b. c. 0 d. 1
8. a. b.
9. a. b. 10
Lesson 10.2
1. 40,320 2. 6,497,400
3. 60 4. 362,880
5. If this were a permutation of 26 letterstaken 3 at a time, you could not use anyletter more than once on a license plate.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.4 127
◆Skill A Finding the probability of mutually exclusive events
Recall Two mutually exclusive events, A and B, cannot both occur at the same time. Insuch a situation the probability of A or B happening is the sum of their individualprobabilities.
◆ ExampleA number cube is tossed. What is the probability of it landing witha. a 4 or 6 on top? b. an even number or 5 on top?c. an even number or a number less than 5 on top?
◆ Solution
a. . Therefore, P(4 or 6) � .
b. . Therefore, P(even or 5) � .
c. These 2 events are not mutually exclusive, because 2 and 4 are even numbersand less than 5. These are inclusive events. Save this problem until youcomplete Skill B for this lesson.
36
�16
�23
P(5) �16
P(even) �36
16
�16
�13
P(6) �16
P(4) �16
Solve the problems that involve mutually exclusive events; giveeach answer as a fraction. Label the others as “inclusive events”and wait to solve them until after you have studied Skill B.
Note: A deck of 52 cards has the following cards.face cards
black
red
1. If there are 270,725 possible 4-card hands, what is the probability thatyou will be dealt a 4-card hand of exactly
a. 4 spades or 4 clubs? b. 4 aces or 4 kings?
c. 4 red cards or 4 black cards? d. 4 face cards or 4 hearts?
2. If there are 2,598,960 possible 5-card hands, what is the probability thatyou will be dealt a 5-card hand of exactly
a. 5 red cards?
b. 5 face cards?
c. 5 red cards or 5 face cards?
d. 5 aces or 5 kings?
e. 5 black cards or 5 diamonds?
A A
22
33
44
55
66
77
88
99
1010
JJ
QQ
KK�hearts
diamonds
A A
22
33
44
55
66
77
88
99
1010
JJ
QQ
KK�spades
clubs
Reteaching
10.4 Using Addition With Probability
Solve each problem.
3. If a number cube is tossed, what is the probability that either an evennumber or a number less than 5 will be on top?
4. Two parts in Exercises 1 and 2 for Skill A of this lesson involved inclusive events. Identify these two Exercises and solve them.
a.
b.
5. A math class of 25 students has 12 boys and 14 students with brown hair. To find the probability of randomly selecting one student that is aboy or has brown hair, what other piece of information would you need to know?
6. A guide shows that out of 60 cable channels available on TV at 8:00 P.M., 8 are premium channels, 6 of which are showing movies. It also showsthat a total of 15 channels are showing movies at this time. If youchannel-surf to a random station, what is the probability that it will be a movie or a premium channel?
7. A surveyor found that in a neighborhood of 150 houses, 55 houses had atleast one dog, 70 houses had a least one cat, and of these houses, 30 had atleast one dog and one cat. What is the probability that a house picked atrandom has
NAME _________________________________________________ CLASS _______________ DATE ______________
128 Reteaching 10.4 Algebra 2
◆Skill B Finding the probability of inclusive events
Recall To find the probability of A or B, when A and B are inclusive events, you mustsubtract the “overlapping” probability. P(A or B) � P(A) � P(B) � P(A and B)
◆ ExampleScott has found some old books in the attic. Out of the 28 books, he notices that8 are novels and 6 are about history. However, 3 out of these 14 books areactually historical novels. If he picks one book to read at random from the 28,what is the probability that it is a novel or history?
◆ Solution
P(novel)
P(history)
P(novel and history)
P(novel or history)
, or approximately 39%
There is a 39% chance that Scott’s choice is a novel or history (or both).
�1128
�828
�628
�328
�3
28
�6
28
�828
Reteaching — Chapter 10
Lesson 10.1
1. 60 2. 12 3. 17,576
4. 362,880 5. 210
6. a. b.
7. a. b. c. 0 d. 1
8. a. b.
9. a. b. 10
Lesson 10.2
1. 40,320 2. 6,497,400
3. 60 4. 362,880
5. If this were a permutation of 26 letterstaken 3 at a time, you could not use anyletter more than once on a license plate.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.5 129
◆Skill A Finding the probability of independent events using individual probabilities
Recall If A and B are independent events, P(A and B) � P(A) � P(B)
◆ Example 1A bag contains 5 blue marbles and 8 red marbles. If you draw 2 marbles from thebag, describe each of the following as dependent or independent events.a. drawing a blue marble, replacing it in the bag, and then drawing a blue marbleb. drawing a blue marble, and without replacing it, drawing another blue marble
◆ Solutiona. Since you replace the first marble drawn, the first draw has no effect on the
second draw. These are independent events.b. The first draw changes the number of marbles in the bag and therefore, the
probability of drawing a blue marble on the second draw. These aredependent events.
◆ Example 2What is the probability of tossing 4 coins such that they all land heads up?
◆ SolutionThe way one coin lands has no effect on the other coins. The 4 landings are
independent events. Since P(heads) for each coin,
. The probability of getting 4 heads is .116
12
�12
�12
�12
�116
�12
Find each probability.
1. A bag contains 8 peppermint and 6 spearmint candies.a. What is the probability of picking a spearmint, replacing it, and then
picking a peppermint?
b. What is the probability of picking a peppermint, replacing it, and then picking a spearmint?
2. The probability that you will get an A in French is 0.75. The probabilityfor getting an A in history is 0.75. What is the probability that you will get an A in both courses?
3. You roll a red number cube and a blue number cube. What is theprobability that you will get
a. a red 5 and a blue 6? b. a red 6 and a blue 5?
c. a total of 11 for both cubes? d. a total of 12 for both cubes?
4. A and B are independent events, P(A) � 0.40, and P(B) � 0.36.
a. Find P(A and B). b. Find P(A or B).
Reteaching
10.5 Independent Events
5. A and B are mutually exclusive events, P(A) � 0.40, and P(B) � 0.36.
NAME _________________________________________________ CLASS _______________ DATE ______________
130 Reteaching 10.5 Algebra 2
◆Skill B Finding the probability of independent events using the complement
Recall The probability of an event that is certain to happen is 1, or 100%. However, for an uncertain event A, it must either happen or not happen. This means that P(A) � P(not A) � 1, or P(not A) � 1 � P(A).
◆ ExampleIf 4 coins are tossed, what is the probability that 0, 1, 2, or 3 coins will land heads up?
◆ SolutionThis includes all possibilities except that of having 4 heads. Since the probability
of all 4 landing heads up is (see Skill A, example 2), the probability of 0, 1, 2,
or 3 heads is .1 �116
�1516
116
Find each probability.
6. Smith and Jones work in the same office. The probability that Smith is onthe phone is 0.5, while the probability that Jones is on the phone is 0.75.Assuming they are not talking to each other, what is the probability at anygiven time that
a. both Smith and Jones are on the phone?
b. Smith or Jones is on the phone?
c. neither is on the phone?
d. Smith is on the phone but Jones is not?
e. Jones is on the phone but Smith is not?
f. only one of them is on the phone?
7. The next 3 batters have batting averages of 0.275, 0.215, and 0.305respectively. What is the probability that
a. all 3 will get a hit? b. none of them gets a hit?
8. The weather report says that the probability of rain is 30% on Mondayand 50% on Tuesday. If these are independent events, what is theprobability that it will rain
a. both days? b. neither day?
c. Monday, but not Tuesday? d. Tuesday, but not Monday?
Reteaching — Chapter 10
Lesson 10.1
1. 60 2. 12 3. 17,576
4. 362,880 5. 210
6. a. b.
7. a. b. c. 0 d. 1
8. a. b.
9. a. b. 10
Lesson 10.2
1. 40,320 2. 6,497,400
3. 60 4. 362,880
5. If this were a permutation of 26 letterstaken 3 at a time, you could not use anyletter more than once on a license plate.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.6 131
◆Skill Finding conditional probability
Recall The probability of two dependent events is called conditional probability.means the probability of event B happening if event A has already occurred.
◆ Example 1Suppose that 65% of the students in the junior class are taking chemistry, 40%are taking German, and 20% are taking both of these courses. What is theprobability that a student selected at randoma. is taking chemistry, if it is known this student is taking German?b. is taking German, if it is known this student is taking chemistry?
◆ SolutionLet C be the event “taking chemistry” and G be the event “taking German.”
a.
If a student is in German, there is a 50% chance this student is also in chemistry.
b.
If a student is in chemistry, there is a 31% chance this student is also in German.
Recall If events A and B are independent, P(A and B) � P(A) � P(B).But, if events A and B are dependent, P(A and B) � P(A) � P( ), which is just
another form of .
◆ Example 2A drawer contains 6 brown, 8 blue, and 4 tan single socks. If you randomly pullout 2 socks, what is the probability that both socks are blue?
◆ SolutionLet F be the event “first sock is blue” and S be the event “second sock is blue.”The probability that both socks are blue is P(F and S) � P(F) � P( ), where
. If the first sock is blue, there are only 7 blue socks left in a drawer of
17 socks, so .
P(F and S) � P(F) � P( )
� 18.3%
�8
18�
717
S�F
P(S�F) �7
17
P(F) �8
18
S�F
P(B�A) �P (A and B)
P(A)
B�A
�0.200.65
� 31%
P(G�C) �P (C and G)
P(C)
�0.200.40
� 50%
P(C�G) �P (C and G)
P(G)
P(B�A) �P (A and B)
P (A)
P(B�A)
Reteaching
10.6 Conditional Probability
Find each probability.
1. The probability that the stock market will go up on Wednesday is 0.25. Ifit goes up on Wednesday, the probability it will go up on Thursday is 0.6.Find the probability the stock market will go up on both Wednesday and Thursday.
2. Recall that P(A or B) � P(A) � P(B) � P(A and B). In Exercise 1, what is the probability that the stock market will go up on Wednesday or Thursday?
3. A survey of a city’s high school students showed that 65% liked the sportsprogram, 40% liked the general activities program, and 25% liked both.What is the probability that a student selected at random from this highschool
a. likes the general activities program if it is known the student likes the sports program?
b. likes the sports program if it is known the student likes the general activities program?
c. likes the sports program or the general activities program?
4. Suppose that P(A) � 0.25 and P( ) � 0.55. Find P(A and B).
5. Suppose that P(A and B) � 0.25, P(A) � 0.4, and P(B) � 0.6.
a. Find P( ). b. Find P(A|B).
6. Suppose a survey of 1500 accidents produces the data in the above table.The table shows the cause of the accident and the gender of the driver ofthe car. Answer these questions based on these data.
a. If an accident is due to poor judgement, what is the probability that the driver is male?
b. If the driver of a car involved in an accident is female, what is the probability that the cause is mechanical failure?
7. There are 6 blue and 4 red marbles in a bag. If you draw two marbles atrandom, what is the probability that they are both blue
a. if you replace the first marble before you draw again?
b. if you do not replace the first marble before you draw again?
8. a. As a review problem, find .
b. The answer to part a should be the same as the answer to part b ofExercise 7. Explain why.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 10.7 133
◆Skill Using a simulation to estimate the experimental probability of an event
Recall Simulations are often used when direct experimentation is very expensive or evenimpossible.
◆ ExampleThe probabilities that a real-estate agent will sell0 to 5 properties in one week are given in thetable at right. a. Distribute 2-digit random integers from 01 to
100 among these values so that thecorresponding random integers can be used to simulate the number of houses thereal-estate agent sells per week.
b. Use this distribution to simulate the agent’sweekly sales for 26 consecutive weeks.
◆ Solutiona. Add a “random numbers” column
to the table.
A probability of 0.12 means thatthere are 12 chances out of 100that this event will happen. Soassign the numbers 1–12 for theevent of selling 0 houses.To follow this pattern, assign thenext 21 numbers for theprobability of 0.21. This will bethe numbers 13–33.Continue so that all numbers from 1 to 100 are assigned.
b. Use your calculator to generate 6 randomnumbers between 1 and 100.
[Your calculator might use a differentcommand from the one shown here,such as Int(rand*100)+1.]
The screen at right shows a practice runfor the first 6 random numbers. Matcheach of these numbers to the appropriate row of the “random numbers” column of the table.
For example, since 46 fits in the 34–59 row, this implies that 2 houses weresold that week.
Verify that the next 5 random numbers shown would imply weekly sales of 4, 4, 1, 3, and 2 houses.
To complete this problem you would need to generate 20 more randomnumbers and match them to the appropriate rows of the table.
Reteaching
10.7 Experimental Probability and Simulation
Number Probability
0 0.12
1 0.21
2 0.26
3 0.19
4 0.12
5 0.10
Total 1.00
Properties sold
Number Probability Randomnumbers
0 0.12 01–12
1 0.21 13–33
2 0.26 34–59
3 0.19 60–78
4 0.12 79–90
5 0.10 91–100
Use a random number generator to complete the followingexercises.
1. A baseball player’s statistics are shown in this table.Complete the last column by assigning randomnumbers from 001 to 1000.
2. Simulate the player’s activity for 10 times at batand record your results.
The probabilities that there are 0 to 4 significant leaksof toxins into a large bay on any one day aregiven in the table at right.
3. Distribute random integers from 0001 to 10,000among these 5 values so that the correspondingrandom integers can be used to simulate the numberof significant toxic spills into the bay each day.
4. Use the distribution you constructed in Exercise 3 to simulate the numberof spills into the bay for 30 consecutive days.
5. How do the probabilities from your simulation compare with theprobabilities given?
◆ SolutionIt would take a long time to write out and find the sum of 100 terms. Here’s a shortcut.
Let tk be the kth term and notice the following pattern.t1 � 1, t100 � 100 and so t1 � t100 � 101t2 � 2, t99 � 99 and so t2 � t99 � 101t3 � 3, t98 � 98 and so t3 � t98 � 101…t50 � 50, t51 � 51 and so t50 � t51 � 101
Since each pair of terms has a sum of 101 and there are 50 of these pairs, the total must be 101 · 50 or 5050.
There is a formula that can be used for this type of problem.
In this example, .�100
k�1k �
100(101)2
� 5050
�n
k�1k �
n(n � 1)2
�100
k�1k
�4
k�1(2k � 1)
�7
k�32k
�
5. a. 0b. 0.76
6. a. 0.375b. 0.875c. 0.125d. 0.125e. 0.375f. 0.5
7. a. � 1.8%b. � 39.6%
8. a. 15%b. 35%c. 15%d. 35%
Lesson 10.6
1. 0.15 2. 0.7
3. a. � 0.38b. 0.625c. 0.79
4. 0.1375
5. a. 0.625b. � 0.417
6. a. � 0.528b. � 0.544
7. a. 0.36
b.
8. a.
b. 10C2 represents the number of ways todraw 2 marbles and 6C2 represents thenumber of ways to draw 2 blue marbles.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 11.2 137
◆Skill A Recognizing arithmetic sequences and finding the common difference
Recall An arithmetic sequence is a sequence in which each pair of successive terms differby the same number d, called the common difference.
◆ Example Based on the terms given, determine whether each sequence is arithmetic. If so,identify the common difference, d.a. 1, 4, 9, 16, 25, . . . b. 8, 5, 2, �1, �4, . . .
Since the differences are not the same, this is not an arithmetic sequence.
b. 5 � 8 � �3 2 � 5 � �3 �1 � 2 � �3 �4 � (�1) � �3Since all the differences are the same this is an arithmetic sequence.The common difference is �3; d � �3.
◆Skill B Finding the nth term of an arithmetic sequence
Recall The nth term, tn, of an arithmetic sequence is given by .
◆ Example 1Find the fifteenth term of the arithmetic sequence that begins with 5, 13, 21, 29,37, . . .
◆ SolutionSince the first term is 5, .
Since 13 � 5 � 8 and 21 � 13 � 8, the common difference is 8, d � 8
The fifteenth term is 117.t15 � 5 � (15 � 1)(8) � 117
t1 � 5
tn � t1 � (n � 1)d
Based on the terms given, state whether or not each sequence isarithmetic. If it is, identify the common difference, d.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 11.4 141
◆Skill A Recognizing geometric sequences and finding the common ratio
Recall A geometric sequence is a sequence in which the ratio of each pair of successiveterms is the same number r, called the constant ratio.
◆ ExampleBased on the terms given, determine whether each sequence is geometric. If so,identify the common ratio, r.a. 3, 6, 12, 24, . . . b. 2, 4, 12, 48, . . . c. 125, 25, 5, 1, . . .
◆ Solution
a.
Since the ratios are the same, this is a geometric sequence.The common ratio is 2; r � 2.
b.
Since the ratios are not the same, this is not a geometric sequence.
c.
Since the ratios are the same, this is a geometric sequence.
The common ratio is .15
; r �15
15
525
�15
25125
�15
4812
� 4124
� 342
� 2
2412
� 2126
� 263
� 2
Determine whether each sequence is a geometric sequence. If so,identify the common ratio, r.
NAME _________________________________________________ CLASS _______________ DATE ______________
144 Reteaching 11.5 Algebra 2
◆Skill C Using mathematical induction to prove a statement
Recall Mathematical induction is used to prove that a statement in terms of n, where n is anatural number, is true for all values of n.
◆ Example Prove by mathematical induction that .
◆ SolutionStep 1: Show that this statement is true when .
Step 2: Let k be some particular value of n and assume that the following is true:
Replace n with k.
Step 3: Write a statement for k � 1 terms. This is the statement to be proved.
Rewrite the right-hand side of this last statement as
Step 4: Recall what was assumed to be true in Step 2, and add 3(k � 1) to eachside of that equation.
common denominator
By showing that the results from steps 3 and 4 (boxed expressions) are the same,you have proved that if the formula is true for k (what you assumed to be true),it is also true for k � 1. Since k is just some value of n, let k � n � 1. Step 1showed that this formula is true when n � 1, and step 2 proved that it must alsobe true for n � 2. If it is true when n � 2, it must also be true for n � 3. Followingthe same logic, it must be true for the next value of n, and eventually for everyvalue of n.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 11.7 147
◆Skill A Finding entries in Pascal’s triangle
Recall Each entry in Pascal’s triangle is the sum of the two entries directly above it.
◆ Example 1Without looking in the textbook, complete the fifth row of Pascal’s triangle.
1 ← Row 01 1
1 2 11 3 3 1
1 4 6 4 1 ← Row 4
◆ SolutionAdding adjacent numbers in row 4 gives 1, 5, 10, 10, 5, 1 for Row 5.
Recall To find the combination of n things taken r at a time, .
◆ Example 2Find the sixth entry in the fifteenth row of Pascal’s triangle.
◆ SolutionInstead of writing out all these rows of the triangle, consider the following.The first entry in any row is 1, which is . To find the rest of the entries, use
.
To find the sixth entry:
The sixth entry in the fifteenth row of Pascal’s triangle is 3003.nCk�1 � 15C6�1 � 15C5 �
15!5!(15 � 5)!
�15!
5!10!� 3003.
nCk�1
nC0
nCr �n!
r!(n � r)!
Complete the following rows for Pascal’s triangle.
1. Row 6:
2. Row 7:
3. Row 8:
Find the value for each entry in Pascal’s triangle.
4. fourth entry in the twelfth row 5. tenth entry in the fifteenth row
6. fifth entry in the eighth row 7. second entry in the eighteenth row
8. eighteenth entry in the eighteenth row 9. tenth entry in the twenty-fifth row
Reteaching
11.7 Pascal’s Triangle
A fair coin is tossed 8 times. Find the following probabilities.
NAME _________________________________________________ CLASS _______________ DATE ______________
148 Reteaching 11.7 Algebra 2
◆Skill B Using Pascal’s triangle to find probabilities
Recall If a probability experiment that has 2 equally likely outcomes is repeated for n
trials, the probability of either outcome occurring exactly k times is .
◆ Example 1A lawyer has two phone lines, A and B. Suppose an incoming call is equally likelyto be on line A or line B. Use Pascal’s triangle to find the probability that 0, 1, 2,3, 4, and 5 out of the next 5 calls will be on line A.
◆ SolutionRow 5: 5C0 5C1 5C2 5C3 5C4 5C5
1 5 10 10 5 1
total: 1 � 5 � 10 � 10 � 5 � 1 � 32
Notice that 32 is 25. (the total of the 5th row entries)
◆ Example 2For the lawyer in Example 1, find the probability that at least 3 of the next 7 callsare on line B.
◆ SolutionAt least 3 means 3, 4, 5, 6, or 7 calls.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 11.8 149
◆Skill A Using the Binomial Theorem to expand binomials raised to a power
Recall ,where
◆ Example 1Write in expanded form.
◆ Solution
Remember that means 5C0. 5C0 � 1
Notice that the powers of a go from 5 to 0 and the powers of b go from 0 to 5.Also, the coefficients are the fifth row of Pascal’s triangle.
◆ Example 2Write 5 in expanded form.
◆ SolutionThis can be done in the same way as Example 1 by replacing a with 2x and b with �y. However, the 2 in the 2x will cause the coefficients to be differentfrom those above.
NAME _________________________________________________ CLASS _______________ DATE ______________
150 Reteaching 11.8 Algebra 2
◆Skill B Writing a particular term of a binomial expansion
Recall Since the first term is found with k � 0, the 6th term is found with k � 6 � 1 � 5.The value of k is always 1 less than the number of the term.
◆ Example 1Write the fourth term of .
◆ Solutionn � 8 and k � 4 � 1 � 3
◆ Example 2Write the term in the expansion of that contains a4b5.
◆ Solutionn � 9 and k � 5 (exponent of b factor) (9
5)a4b5 � 126a4b5
(a � b)9
(83)x5(�3y)3 � �1512x5y3
(x � 3y)8
◆Skill C Using the Binomial Theorem to determine probabilities
Recall If a probability experiment has two possible outcomes that are not equally likely,the sum of the two probabilities must still equal 1.
◆ ExampleThe probability that a movie is showing on any one of a block of 5 premiumcable channels is 80%. What is the probability that at any given time 4 of the 5channels are showing a movie?
◆ SolutionFor each channel, P(showing movie) � 0.8 and P(not showing movie) � 0.2.
There is approximately a 41% chance that exactly 4 movies are showing.
(51)(0.8)4(0.2)1 � 0.41
Ten students assume that the chance of getting an A on aparticular chemistry test is 75%. If this is true, what is theprobability that
12.2 Stem-and-Leaf Plots, Histograms, and Circle Graphs
b. You should list the leaves for each stemfrom least to greatest. Because there are 16data items, the median is the average ofthe eighth and ninth items counting fromthe least value.
The median is 20.
The mode is 10 because this numberoccurs most often.
18 � 222
� 20
◆Skill B Making a histogram
Recall The height of each vertical bar in a histogram represents the frequency of the valueor values marked below it on the horizontal axis.
◆ ExampleThis set of data represents shoe sizes for 9 9.5 10 8.5 9 11a group of boys. 9 9.5 10.5 10 11 8
Make a frequency table and a histogram for this set of data.
◆ Solution
shoe sizes8 8.5 9 9.5 10 10.5 11
3
2
1
num
ber
of
bo
ys
size 8 8.5 9 9.5 10 10.5 11
frequency 1 1 3 2 2 1 2
On your own paper, make a frequency table and histogram foreach data set.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 12.3 155
◆Skill A Finding the range, quartiles, and interquartile range
Recall The range is the difference between the maximum and minimum values.The quartiles divide the data into 4 sets where Q2 is the median.The interquartile range is given by .
◆ Example Earthquake intensities are measured on the Richter scale. For the earthquake intensities listed at right, find the following:a. rangeb. the quartilesc. the interquartile range
◆ Solutiona. Arrange the data in order from least to greatest.
b. Find the median of all the data values and label it Q2.Find the median of the lower half of the values and label it Q1.Find the median of the upper half of the values and label it Q3.
4. The lengths of several samples of beetles in centimeters are: 2.5, 2.8, 3.1,3.6, 3.4, 3.8, 3.0, 2.8, 3.5, 3.3, 2.6, 3.0, 2.9, 2.7, 3.4, 3.2, 3.7, 2.5, 3.1, 2.9,2.5, 3.1, 3.8, 2.9
5. The ages of the members of the Richards family reunion are: 4, 7, 9, 31,34, 2, 11, 33, 36, 2, 8, 13, 35, 37, 24, 34, 31, 50, 52, 57, 60, 69, 78, 83.
6. Number of students over 4 weeks needing before-school tutoring inalgebra: 7, 12, 10, 8, 7, 12, 15, 10, 10, 7, 2, 9, 12, 14, 6, 10, 12, 9, 8, 8.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 12.4 157
◆Skill A Finding the range and the mean deviation
Recall The sum 1 � 2 � 3 � 4 can be represented in summation notation as .
gives the mean deviation of a set of data.
◆ ExamplePaul’s test grades are 78, 80, 75, 85, and 82; Marie’s test grades are 60, 83, 73, 100,and 84. Compare the range and mean deviation for these two sets of test grades.
◆ SolutionPaul’s range of test grades is 85 � 75 � 10; Marie’s range is 100 � 60 � 40.The range, which measures the spread of values, is much greater for Marie’s grades.
To find the mean deviation, you must first find the mean.
Paul: Marie:
Then find the absolute value of the difference between the mean, , and each test grade.
Paul: Marie:
Although Paul and Marie have the same mean grade, the mean deviation showsthat the dispersion of Marie’s grades is much greater than Paul’s.
� 10.8� 2.8
15 �
5
i�1�xi � 80� �
15
� 5415 �
5
i�1�xi � 80� �
15
� 14
�5
i�1�xi � 80� � 54�
5
i�1�xi � 80� � 14
x
x �60 � 83 � 73 � 100 � 84
5� 80x �
78 � 80 � 75 � 85 � 825
� 80
1n �
n
i�1�xi � x�
�4
i�1i
Find the range and mean deviation for each data set.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 12.5 159
◆Skill A Finding the probability of exactly r successes in n trials of a binomial experiment
Recall If the probability of success in a binomial experiment is p, then the probability ofexactly r successes in n trials is given by .
◆ ExampleIf your probability of receiving e-mail is 40% each day, what is the probabilitythat you will receive e-mail exactly 5 out of the next 7 days?
◆ Solutionprobability of success: p � 0.4probability of failure: 1 � p � 0.6
Use a calculator.
The probability of receiving e-mail exactly 5 out of the next 7 days is about 7.7%.
� 0.077� 7C5(0.4)5(0.6)2
P � nCrpr(1 � p)n�r
r � 5n � 7
P � nCrpr(1 � p)n�r
Suppose that the probability of rain is 60% for each of the next 5days. Find the probability that it will rain on
1. exactly 2 days. 2. exactly 3 days.
3. all 5 days. 4. none of these days.
Reteaching
12.5 Binomial Distributions
◆Skill B Finding the probability of exactly r successes or s successes in n trials of a binomialexperiment
Recall Since exactly r successes and exactly s successes are mutually exclusive events, theprobability of r successes or s successes is the sum of these two separate probabilities.
◆ ExampleTheresa has calculated that her probability of getting a strike in any one frame of a bowling game is 30%. What is the probability that she will get a strike in exactly 3 or 4 frames out of 10 in her next game?
◆ Solution
The probability that Theresa will get exactly 3 or 4 strikes is approximately 46.7%.
� 0.467� 10C3(0.3)3(0.7)7 � 10C4(0.3)4(0.7)6
� nCs ps(1 � p)n�sP � nCrpr(1 � p)n�r
s � 4r � 3n � 101 � p � 0.7p � 0.3
The probability that you will have to stop for a red light at thelast intersection before you reach school is 80%. In the next 10times you drive up to this intersection, what is the probabilitythat the light will be red
5. exactly 5 or 6 times? 6. exactly 5, 6, or 7 times?
NAME _________________________________________________ CLASS _______________ DATE ______________
160 Reteaching 12.5 Algebra 2
◆Skill C Finding the probability of “at least” r successes in n trials or “at most” r successes inn trials of a binomial experiment
Recall At most means less than or equal; at least means greater than or equal.
◆ ExampleThe owner of a local bookstore found that 64% of all the books she sold werefiction. Find the probability that out of the next 5 books she sellsa. at least 3 are fiction. b. at most 2 are fiction.
◆ Solution
a. “At least 3” means 3, 4, or 5.
The probability that at least 3 of the next 5 books will be fiction isapproximately 74.9%.
b. “At most 2” means 0, 1, or 2.
The probability that at most 2 of the next 5 books will be fiction isapproximately 25.1%.
Notice that the answers to parts a and b have a sum of 1. This is true because “at least 3” and “at most 2” cover all the possibilities.
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 12.6 161
◆Skill A Finding the probability of an event using a normal distribution curve
Recall For data that is normally distributedwith mean and standard deviation�, the normal distribution curve atright gives the approximate percent ofdata that will fall within the unitsmarked on the horizontal axis.
◆ ExampleThe scores on a college entrance exam for graduates of a particular high school are normally distributed with a mean of 580 and a standard deviation of 120. Find the probability that a randomly selected student from this high school scoreda. between 460 and 700. b. over 820.
◆ Solutiona. 460 � 580 � 120, so 460 .
According to the curve above, the percent of scores between and is 34%.700 � 580 � 120, so 700 .The percent of scores between and is 34%.
34% � 34% � 68%; the probability of scoring between 460 and 700 is about 68%.
b. 820 � 580 � 2(120), so 820 .
The percent of scores above is 2%.Thus, the probability of scoring over 820 is approximately 2%.
x � 2�
� x � 2�
x � �x� x � �
xx � �
� x � �
x
Use the normal distribution curve shown above to complete thefollowing exercises.
1. The mean for a set of scores is 100 and the standard deviation is 7.5. Findthe probability that a randomly selected score is
a. between 85 and 100. c. less than 100.
b. between 85 and 115.
2. The average age of the employees in a large company is 38 with a standarddeviation of 8. If an employee is selected at random, what is theprobability that the employee’s age is
a. between 38 and 54? c. more than 46?
b. less than 22?
Reteaching
12.6 Normal Distributions
x – 3� x – 2� x – � x x + � x + 2� x + 3�
2%14%
34% 34%14%
0.2
2%
Use a graphics calculator to complete the following exercises.
3. An auto parts manufacturer finds that the useful life of a spark plug in acar is approximately normally distributed with a mean of 9 months and astandard deviation of 2.3 months. What is the probability that arandomly selected spark plug will last
a. more than 1 year? b. less than 7 months?
4. Mortgage statistics collected by a bank indicate that the number of years the average new homeowner will occupy a house before moving or selling is normally distributed with a mean of 6.3 years and a standard deviation of 2.31 years. What is the probability that a randomly selected homeowner
NAME _________________________________________________ CLASS _______________ DATE ______________
162 Reteaching 12.6 Algebra 2
◆Skill B Finding the probability of an event using z-scores
Recall z-scores tell how many standard deviations a data value is above or below the mean.
, where a is a given data value.
You will need a graphics calculator for the exercises on this page.
◆ ExampleThe mean shelf life of a particular dairy product is approximately 11 days with astandard deviation of 3 days. If one of these dairy products is randomly selected,find the probability that it will lasta. between 10 and 15 days. b. less than 7 days.
◆ Solutiona. When a � 10, or approximately �0.33.
When a � 15, or approximately 1.33.
Use a calculator to graph
in the window
indicated at right. Find the shadedarea by using the feature(with a lower limit of �0.33 and aupper limit of 1.33) or a similarfeature on the calculator you areusing.
Since the shaded area in this example is approximately 0.54, the probability ofthis dairy product lasting between 10 and 15 days is approximately 54%.
b. When a � 7, , or approximately �1.33.
Use your calculator to find the area from x � �4 to x � �1.33.
The probability of the dairy product lasting less than 7 days is about 0.09, or 9%.
The reference angle is 65°. The reference angle is 30°. The reference angle is 45°.
225� � 180� � 45�
115°
x
y
ref.
x
y
60°
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 13.2 165
◆Skill A Finding coterminal angles
Recall An angle is in standard position if its vertex is at the origin and its initial side isalong the positive x-axis. Angles in standard position are coterminal if they havethe same terminal side.
◆ Example Which of 120°, 420°, and �300°are coterminal with 60°?
◆ SolutionThe measures of coterminal angles differ by a multiple of 360°.120° � 60° � 60° not a multiple of 360°420° � 60° � 360° is a multiple of 360°�300° � 60° � �360° is a multiple of 360°The angles coterminal with 60° are 420° and �300°.
For � in standard position, select the listed angles that arecoterminal with �.
Find each trigonometric value. Give exact answers.
4. sin 120° 5. cos 330° 6. tan 225° 7. cos 150°
8. sin 240° 9. sin 150° 10. tan 315° 11. cos 225°
NAME _________________________________________________ CLASS _______________ DATE ______________
168 Reteaching 13.3 Algebra 2
A reference angle is the positive acute anglebetween the terminal side of a given angleand the x-axis.
One mnemonic for remembering whichfunctions are positive in each quadrant is“All students take calculus.”
◆ Example Find each exact value.a. sin 315° b. cos 240° c. tan 210°
◆ Solutiona. 315° is in Quadrant IV, b. 240° is in Quadrant III, c. 210° is in Quadrant III,
where sine is negative. where cosine is where tangent is The reference angle is negative. The reference positive. The reference45°. angle is 60°. angle is 30°.sin 315° � �sin 45° cos 240° � �cos 60° tan 210° � tan 30°
◆Skill C Finding the coordinates of a point P on a circle
Recall If P(x, y) lies at the intersection of the terminal side of � in standard position and acircle centered at the origin with radius r, then P(x, y) � P(r cos �, r sin �).
◆ Example Find the coordinates of point P shown inthe figure at right.
◆ Solutionand
and
The coordinates of point P are .(�2�3, � 2)
r sin � � 4(�12)r cos � � 4(�
�32 )
sin 210� � �sin 30� � �12
cos 210� � �cos 30� � ��32
x
y
210°
r = 4
P(x, y)
ref.
Point P is located at the intersection of a circle centered at the originwith a radius of r and the terminal side of angle � in standard position.Find the exact coordinates of point P.
12. � � 135°, r � 6 13. � � 30°, r � 10 14. � � 300°, r � 12
Find the measure of each angle to the nearest whole degree.
13. Find the measure of the smallest angle in a right triangle with sides of 3, 4, and 5 centimeters.
14. What is the angle between the bottom of theladder and the ground as shown at right?
15. Find the angle at the peak of the roof as shownat right.
16. The hypotenuse of a right triangle is 3 times as long as the shorter leg.Find the measure of the angle between the shorter leg and thehypotenuse.
NAME _________________________________________________ CLASS _______________ DATE ______________
174 Reteaching 13.6 Algebra 2
◆ Skill B Applying inverse trigonometric functions
Recall
◆ ExampleAt a certain time of the day, the 5 meter flagpoleshown at right casts a shadow that is 3 meters long.What is the angle of elevation of the sun at this time?
◆ SolutionSince 3 meters is the length of the side adjacent to �and 5 meters is the length of the side opposite �, usethe tangent function.
This last equation states that � is the angle that has a tangent of .
NAME _________________________________________________ CLASS _______________ DATE ______________
180 Reteaching 14.3 Algebra 2
◆Skill B Simplifying expressions by using basic trigonometric identities
Recall Since sin2 � � cos2 � � 1, then sin2 � � 1 � cos2 � and cos2 � � 1 � sin2 �.
◆ Example 1Simplify csc � tan � to sec �.
◆ Solutioncsc � tan � fundamental identities
fundamental identity
◆ Example 2Simplify (sec � � 1)(sec � � 1) to tan2 �.
◆ Solution(sec � � 1)(sec � � 1) � sec2 � � 1
� tan2 � � 1 � 1 fundamental identity� tan2 �
(a � b)(a � b) � a2 � b2
� sec �
sin �sin �
� 1�1
cos �
�1
sin ��
sin �cos �
◆Skill C Finding the values for which a trigonometric expression is defined
Recall Division by zero is undefined. sin � and cos � are defined for all values of �.tan � and sec � are defined for all values of � except odd multiples of 90�.cot � and csc � are defined for all values of � except multiples of 180�.
◆ ExampleFor what values of � is defined?
◆ Solutionsin � � 0, if � is a multiple of 180�.Since these values of � would give a denominator of 0 in the given expression,
is defined for all values � except � � n180� (any multiple of 180�).1 � cos �sin �
1 � cos �sin �
Find the values of � for which each trigonometric expression isdefined.
11. 12. 13. sin �sin2 � � cos2 �
1 � cos �tan �
sin �cos2 � � 1
7. 8. 9. 10. 90� 11. 0�
12. 90� 13. 37� 14. 76� 15. 113�
16. 71�
Reteaching — Chapter 14
Lesson 14.1
1. 5.4 square centimeters
2. 92.2 square kilometers
3. 14.7 square meters
4. 30 square feet
5.
6.
7.
8. 2 triangles
9. no triangles
10. 2 triangles
11. 1 triangle
Lesson 14.2
1.
2.
3.
4.
Answers may vary due to rounding.
5.
6.
7.
8.
Lesson 14.3
1.
2.
3.
4.
5. sin x cot x � sin x � cos x
6. sin x sec x cot x � sin x � 1
7. cos2 x � sin2 x � 1 � sin2 x � sin2 x �1 � 2 sin2 x
NAME _________________________________________________ CLASS _______________ DATE ______________
Algebra 2 Reteaching 14.4 181
◆Skill A Evaluating expressions using the sum and difference identities
Recall sin(A � B) � sin A cos B � cos A sin Bsin(A � B) � sin A cos B � cos A sin Bcos(A � B) � cos A cos B � sin A sin Bcos(A � B) � cos A cos B � sin A sin B
Note: You may want to review finding the values of trigonometric functions ofcommon angles in Lesson 13.3 before completing the following examplesand exercises.
◆ ExampleFind an exact value for
a. . b. .
◆ Solution
a.
Notice that .
b.
Notice that .π3
�π6
�π2
and cos π2
� 0
� 0
��34
��34
�12
��32
��32
�12
cos(π3
�π6) � cos π
3 cos π
6� sin π
3 sin π
6
sin(π3
�3π4 ) sin π
3� sin 3π
4
��2 � �6
4
� ��64
��24
��32
� ��22
�12
��22
sin(π3
�3π4 ) � sin π
3 cos π
4� cos π
3 sin 3π
4
cos(π3
�π6)sin(π
3�
3π4 )
Find the exact value of each expression.
1. 2.
3. 4. cos(3π4
�π2)cos(2π
3� π)
sin(3π4
�π6)sin(π
4�
π3)
Reteaching
14.4 Sum and Difference Identities
Find the image of each point after the given counterclockwiserotation.
NAME _________________________________________________ CLASS _______________ DATE ______________
182 Reteaching 14.4 Algebra 2
◆Skill B Using a matrix to find the coordinates of an image point after a rotation of � degrees
Recall If P(x, y) is any point in the plane, then the image of P(x, y) after a rotation of �degrees counterclockwise about the origin is given by the following matrix product.
◆ ExampleFind the image of (3, 4) after a rotation of 120� counterclockwise about the origin.
◆ Solution
�
In your calculator, let and .
The product [A][B] is approximately .
The coordinates of the image point are approximately (�4.96, 0.60).
��4.960.60�
[B] � �34�� �
12
�32
��32
�12�[A] �
� �12
�32
��32
�12� � �3
4��cos 120�sin 120�
�sin 120�cos 120�� � �3
4�
x
y
O
(3, 4)120°
(–4.96 , 0.60)
�cos �sin �
�sin �cos �� � �x
y�
9. tan x � cot x �
sec x csc x
10. (cos x � sin x)2 � cos2 x � 2 cos x sin x �sin2 x � 1 � 2 cos x sin x