Vocabulary Builder - Salamanca High School€¦ · 4-1 Quadratic Functions and Transformations Review 1. Circle the vertex of each absolute value graph. Vocabulary Builder parabola
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Got It? The graph shows the jump of a dolphin. The axis of symmetry is x 5 2, and the height is 7. If the path of the jump passes through the point (5, 5), what quadratic function models the path of the jump?
15. The vertex is Q , R .
16. Substitute h and k in the vertex form f (x) 5 a(x 2 h)2 1 k.
y 5 aQx 2 R21
Problem 5
Using Vertex Form
Got It? What is the graph of f (x) 5 2(x 1 2)2 2 5?
13. Multiple Choice What steps transform the graph of y 5 x2 to y 5 2(x 1 2)2 2 5?
14. Circle the graph of f(x) 5 2(x 1 2)2 2 5.
Problem 4
12. Use one of the functions below to label each graph.
y 5 (x 1 3)2 y 5 x2 2 1 y 5 (x 2 2)2 1 3 y 5 (x 1 1)2 2 2
y
x
yx
y
x
yx
y
x
y
x
y
x
yx
0 2 4 6 8
8
6
4
2
0
y
x
Refl ect across the x-axis, stretch by the factor 2, and translate 2 units to the left and 5 units up.
Stretch by the factor 2 and translate 2 units to the right and 5 units up.
Stretch by the factor 2 and translate 2 units to the left and 5 units down.
Refl ect across the x-axis, stretch by the factor 2, and translate 2 units to the left and 5 units down.
Got It? The Zhaozhou Bridge in China is the oldest known arch bridge, dating to 605 a.d. You can model the support arch with the function f (x) 5 20.001075x2 1 0.131148x, where x and y are measured in feet. How high is the arch above its supports?
20. What point on the parabola gives the height of the arch above its supports?
Main Idea: Modeling is a way of using math to describe a real-world situation.
Definition: A function or equation models an action or relationship by describing its behavior or the data associated with that relationship.
Example: The equation a 5 3g models the relationship between the number of apples, a, and the number of oranges, g, when the number of apples is triple the number of oranges.
Use Your Vocabulary
Draw a line from each description in Column A to the equation that models it in Column B.
Column A Column B
2. The string section of the orchestra has twice y 5 2x 1 1as many violins as cellos.
3. There are two eggs per person with one y 5 100 2 2xextra for good measure.
4. There were 100 shin guards in the closet, and y 5 2xeach player took two.
Got It? What is the equation of a parabola containing the points (0, 0), (1, 22), and (21, 24)?
5. Substitute the three points one at a time into y 5 ax2 1 bx 1 c to write a system of equations.
Use (0, 0). 5 aQ R21 bQ R 1 c
Use (1, 22). 5 aQ R21 bQ R 1 c
Use (21, 24). 5 aQ R21 bQ R 1 c
6. Solve the system of equations.
7. The equation of the parabola is y 5 x2 1 x 1 .
Problem 1
Using a Quadratic Model
Got It? The parabolic path of a thrown ball can be modeled by the table. The top of a wall is at (5, 6). Will the ball go over the wall? If not, will it hit the wall on the way up, or the way down?
8. Circle the system of equations you find by substituting the three given points that are on the parabola.
1 5 9a 1 3b 1 c 3 5 a 1 b 1 c 3 5 a 1 b 1 c 2 5 25a 1 5b 1 c 5 5 2a 1 2b 1 c 5 5 4a 1 2b 1 c 3 5 36a 1 6b 1 c 6 5 9a 2 3b 1 c 6 5 9a 1 3b 1 c
9. Now, solve the system of equations.
10. The solution of the system is a 5 , b 5 , c 5 .
11. The quadratic model for the ball’s path is .
12. How can you determine whether the ball goes over the wall? Place a ✓ if the statement is correct. Place an ✗ if it is not.
Got It? The table shows a meteorologist’s predicted temperatures for a summer day in Denver, Colorado. What is a quadratic model for the data? Predict the high temperature for the day. At what time does the high temperature occur?
15. Using the LIST feature on a graphing calculator, identify the data that you will enter.
L1 5
L2 5
16. Using a 24-hour clock, write the values for the L1 column.
6 a.m.: 3 a.m.:
9 a.m.: 6 p.m.:
12 p.m.: 9 p.m.:
17. Circle the calculator screen that shows the correct data entry.
6912369
L1 L2 L3637686898576
L2(6) 76
6912151821
L1 L2 L3637686898576
L2(6) 76
6912151821
L1 L2 L3768689857663
L2(6) 63
18. Enter the data from the table into your calculator. Use the QuadReg function. Your screen should look like the one at the right.
Error Analysis Your classmate says he can write the equation of a quadratic function that passes through the points (3, 4), (5, 22), and (3, 0). Explain his error.
22. Graph the points (3, 4), (5, 22), and (3, 0).
23. Underline the correct words to complete the rule for finding a quadratic model.
Two / Three noncollinear points, no two / three of which
are in line horizontally / vertically , are on the graph of exactly
Main Idea: The factors of an expression are similar to the factors of a number.
Definition: The factors of a given expression are expressions whose product equals the given expression. When you factor an expression, you break it into smaller expressions whose product equals the given expression.
Example: The factors of the expression 2x2 2 x 2 10 are 2x 2 5 and x 1 2.
Got It? What is the expression x2 2 11x 1 30 in factored form?
6. Underline the correct word(s) to complete each sentence.
I need to find factors that multiply / sum to 30 and multiply / sum to 211.
At least one of the factors that sum to 211 must be positive / negative .
The two factors that multiply to 30 must both be positive / negative .
7. Circle the factors of 30 that sum to 211.
1 and 30 2 and 15 3 and 10 5 and 6
21 and 230 22 and 215 23 and 210 25 and 26
8. Factor the expression.
x2 2 11x 1 30 5 Qx RQx R
Got It? What is the expression 2x2 1 14x 1 32 in factored form?
9. Rewrite the expression to show a trinomial with a leading coefficient 1.
2x2 1 14x 1 32 5
10. Reasoning You are looking for factors of 232 that sum to 214. Which of the factors has the greater absolute value, the negative factor or the positive factor? How do you know?
Reasoning Explain how to rewrite the expression a2 2 2ab 1 b2 2 25 as the product of two trinomial factors. (Hint: Group the first three terms. What type of expression are they?)
21. Complete: The first three terms of the expression are a 9.
perfect square trinomial difference of two squares
22. Factor the first three terms of the expression.
23. Rewrite the original expression using the factored form of the first three terms.
24. Complete: The expression you wrote in Exercise 23 is a 9.
perfect square trinomial difference of two squares
25. Circle the expression written as the product of two trinomial factors.
a2 2 2ab 1 b2 (a 2 b)2 2 25 (a 2 b)(225) (a 2 b 2 5)(a 2 b 1 5)
Check off the vocabulary words that you understand.
factor of an expression perfect square trinomial difference of two squares
Rate how well you can factor quadratic expressions.
19. Use the justifications to complete each step.
64x2 2 16x 1 1 Write the original expression.
Q xR2 2 16x 1 Q R2 Write the first and third terms as squares.
Q xR22 2Q RQ R x 1 Q R2
Write the middle term as (2ac)x.
20. Write the expression as the square of a binomial.
1. Cross out the equation below that is not a function.
f (x) 5 2x 2 7 y2 5 3x2 2 4x y 5 2x2 1 14x 2 7 g(x) 5 u x3 u
Vocabulary Builder
zero of a function (noun) ZEER oh
Main Idea: Wherever the graph of a function y 5 f (x) intersects the x axis, f (x) 5 0. The value of x at any of these intersection points is called a zero of the function.
Definition: A value of x for which f (x) 5 0 is a zero of the function f (x) .
Example: x 5 2 is a zero of f (x) 5 3x 2 6, because f (2) 5 3(2) 2 6 5 0.
Use Your Vocabulary
Write the zero(s) of each function.
2. 3. 4.
y
xO2 2
2
y
xO 2
2
y
xO2 2
2
Zero(s): Zero(s): Zero(s):
If ab 5 0, then a 5 0 or b 5 0.
Example: If (x 1 7)(x 2 2) 5 0, then (x 1 7) 5 0 or (x 2 2) 5 0.
5. If either x 1 7 5 0 or x 2 2 5 0, circle all of the possible values of x.
Got It? What are the solutions of the quadratic equation x2 1 2x 2 24 5 0?
12. The graph at the right shows the equation. Circle the zeros of the function.
13. The solutions of the quadratic equation
are and .
Using a Quadratic Equation
Got It? The function y 5 20.03x2 1 1.60x models the path of a kicked soccer ball. The height is y, the distance is x, and the units are meters. How far does the soccer ball travel? How high does the soccer ball go? Describe a reasonable domain and range for the function.
14. The graph below shows the function. Circle the point on the graph where the soccer ball is at its highest point and the point where the soccer ball lands. Label each point with its coordinates.
Reasoning Circle the phrase that completes each sentence.
15. The distance the soccer ball travels is the 9.
x-coordinate ofthe vertex
y-intercept
x-coordinate of thepositive zero
y-coordinate of the vertex
16. The maximum height of the soccer ball is the 9.
x-coordinate ofthe vertex
y-intercept
x-coordinate of thepositive zero
y-coordinate of the vertex
17. Underline the correct word to complete each sentence.
The domain should include positive / negative numbers only.
The range should include positive / negative numbers only.
Draw a line from each expression to its square root.
1. 25x2 x 1 2
2. x2 1 4x 1 4 6x 2 3
3. 36x2 2 36x 1 9 2x 2 5y
4. 4x2 2 20xy 1 25y2 45x
Vocabulary Builder
trinomial (noun) try NOH mee ul
Related Words: perfect square
Definition: A trinomial is an expression consisting of three terms.
Main Idea: You can use perfect square trinomials to solve quadratic equations.
Examples: 4x2 2 7x 1 5, ax2 1 bx 1 c, and 2x 2 5y 1 4z are all trinomials. x2 1 4x 1 4 is a perfect square trinomial because it is the square of the binomial x 1 2.
Use Your Vocabulary
5. Write the number of terms in each expression.
x 1 1 t 2 2 2t 2 6 y 3 p2 2 6p 1 9
6. Put a T next to each expression that is a trinomial. Put an N next to each expression that is not a trinomial.
x2 g 3 1 g 2 4 x2 2 2x 1 5 x2 2 4x
7. Cross out the expression that is NOT a perfect square trinomial.
Got It? The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 2822.4 in2. What are the window dimensions?
8. Circle the equation that represents this situation.
Got It? What is the number of real solutions of 2x2 2 3x 1 7 5 0?
18. Complete the reasoning model below.
Problem 3
15. Cross out the value that will NOT be substituted into the Quadratic Formula to solve the problem.
21 1 48 2400
16. Substitute values for a, b, and c into the Quadratic Formula and simplify.
x 54" Q R2
2 4Q R Q R2Q R
x 5
4"
< or x <
17. The smallest amount you can charge is for each CD to make a profit
of $100.
WriteThink
a , b , c
b2 4ac 2 4
Find the values of a, b, and c.
Evaluate b2 4ac.
Interpret the discriminant.The discriminant is positive / negative / zero .
The equation has 2 / 1 / 0 real solution(s).
Chapter 4 108
Applying the Quadratic Formula
Got It? Fundraising Your School’s jazz band is selling CDs as a fundraiser. The total profit p depends on the amount x that your band charges for each CD. The equation p 5 2x2 1 48x 2 300 models the profit of the fundraiser. What’s the least amount, in dollars, you can charge for a CD to make a profit of $100?
14. Circle the equation that represents the situation.
Got It? Reasoning You hit a golf ball into the air from a height of 1 in. above the ground with an initial vertical velocity of 85 ft/s. The function h 5 216t2 1 85t 1 1
12 models the height, in feet, of the ball at time t in seconds. Will the ball reach a height of 120 ft? Explain.
1. Circle the square root that is not a real number.
!64 !6 2 (2)(4) !4 2 (2)(26) "(25)2
Vocabulary Builder
conjugate (adjective) KAHN juh gut
Related Words: complex numbers, pairs, roots, imaginary solutions
Math Usage: The conjugate of the complex number a 1 bi is a 2 bi .
Main Idea: Complex solutions occur in conjugate pairs of the form a 1 bi and a 2 bi . The product of complex conjugates is always a real number. You can use complex conjugates to simplify division of complex numbers.
Use Your Vocabulary
Write C if the number pairs are complex conjugate or N if they are not.
2. 4 1 3i, 4 2 3i
3. 5 1 !2, 5 2 !2
4. !5 2 !3i, !5 1 !3i
5. 3 1 !5i, 3 1 !25i
Th e imaginary unit i is the complex number whose square is 21. So, i 2 5 21,
and i 5 !21.
For any positive real number a, !2a 5 !21 ? a 5 !21 ? !a 5 i!a.
Note that A!25 B2 5 Ai!5 B2 5 i 2A!5 B2 5 21 ? 5 5 25 (not 5).
1. The solution of system y 5 3x 1 2 and y 5 5x is the point where the two lines intersect.
2. The solution of a system of 2 linear equations has at most 2 points of intersection.
3. The solution of a system of inequalities is the point where the lines intersect with the y-axis.
4. The solution of a system of inequalities is the region where the graphs of the inequalities overlap.
Vocabulary Builder
Quadratic-Linear System (noun) kwah DRAT ik LIN ee ur SIS tum
Related Words: System of equations, system of inequalities.
Main Idea: A system of equations can include an equation with a graph that is not a line. Such a system can have more than one solution.
Definition: A quadratic-linear system is a system of one quadratic equation and one linear equation. The system can have two, one, or no solutions (points of intersection).
Use Your Vocabulary
5. Cross out the graph that does NOT illustrate a quadratic-linear system.