Entering Algebra 1 Summer Packet 2013

Math Summer Packet For

Students Entering Algebra 1

Math Summer Packet For

Students Entering Algebra 1

Doral Academy Prep School

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Math Department

rents and Students,

This Summer Packet has been designed to provide the students entering Algebra 1 with an appropriate

of basic skills and concepts they have already been exposed to on previous math courses. It is intended

the transition into a new math discipline. It is not intended to be worked on one sitting. This packet

ollected and graded. It will be the first grade you will receive; make it count. You must show all your

hen calculations are necessary to receive credit. Calculator answers are not acceptable.

What is algebra?

Algebra is a foundational math discipline. The skills and concepts you will learn through this course will

again, and again, throughout the rest of your mathematical journey. This includes college.

If you give this course the importance it deserves, the rest of your math courses will become easier to

and. Some of these concepts, like Literal Equations, can also be applied to science (You will see it when

s are used!). Use this class to your advantage!

Familiarize yourself with the Geometry – Algebra 1 EOC Reference sheet. While you will be provided

n class, and you will be allowed to use it during the End Of Course Test, you must know how to use it to

vantage. That includes knowing the meaning of each term included in it.

rtners in education,

Academy’s Math Department Team

3

Vocabulary

Use the glossary uploaded in the

school’s website to complete the

vocabulary exercises included in the

following pages. Although the

glossary is for 9th grade, these terms

you learned through your 8th grade

math class.

4

One-Step Equations

Complete each sentence using a term from the box.

algebraic equivalent graphical

inverse one-step equation solve an equation

1. A(n) ___________________ check of a solution is to substitute an obtained value into an equation to determine if it

produces a true statement.

2. Two equations are ___________________ if they have the same solution or solutions.

3. A(n) ___________________ check of a solution is to use the graph of an equation to determine if a value is a solution

to an equation.

4. Operations that undo each other are known as ___________________ operations.

5. If only one operation is required to solve an equation, then it is a(n) ___________________.

6. To ___________________ means to find values that produce a true statement.

The Coordinate Plane

Match each definition to its corresponding term.

a. Cartesian coordinate systemb. coordinate planec. ordered paird. origin

e. x-axisf. x-coordinateg. y-axish. y-coordinate

____ 7. The horizontal number line in a Cartesian coordinate system

____ 8. A pair of numbers of the form (x,y) that represents a unique position in the coordinate plane

____ 9. The vertical number line in a Cartesian coordinate system

____ 10. The first number in an ordered pair

____ 11. A method of representing the location of a point using an ordered pair of real numbers of the form (x,y)

____ 12. A plane formed by the intersection of a vertical real number line and a horizontal real number line

____ 13. The second number in an ordered pair

____ 14. The point where the x-axis and y-axis intersect in the coordinate plane

5

Relations and Functions


a. dependent variableb. domainc. functiond. independent variablee. input

f. outputg. rangeh. relationi. set notation

____ 15. an indication that a group of numbers is part of a set

____ 16. the variable that represents the input value of a function

____ 17. the set of all output values for a function

____ 18. the variable that represents the output value of a function

____ 19. any set of ordered pairs

____ 20. a relation in which for every input there is exactly one output

____ 21. the set of all input values for a function

____ 22. the second coordinate of an ordered pair in a relation

____ 23. the first coordinate of an ordered pair in a relation

Evaluating Functions, Function Notation, Domain, and Range

Complete each sentence using one of the terms from the box.

domain evaluate a function function

function notation range

24. To _____________________ is to replace the variable with the given value and find the result.

25. A(n) _____________________ is a relation in which every input has exactly one output.

26. A method of writing functions such that the dependent variable is replaced with the name of the function is called

_____________________.

27. The _____________________ is the set of all output values of a function.

28. The _____________________ is the set of all input values of a function.

6

Multiple Representations of Linear Functions

Write the term that best completes each statement.

linear equation linear functions function notation ordered pair

29. A __________________ is an equation whose graph is a straight line.

30. The letters “ f(x)” can be used when a function is written using __________________.

31. An __________________ shows the location of any point on a coordinate grid.

32. All linear equations are also __________________, except when they are in the form x = a, where a represents any

number.

Finding the Slope of a Line

Match each definition or description to its corresponding term.

a. unit rateb. slopec. vertical changed. horizontal change

e. rate of changef. ratiog. positive slopeh. negative slope

____ 33. comparison of two quantities written as a:b or as a fraction

____ 34. describes a line that slants upward from left to right

____ 35. difference in x-coordinates when computing slope

____ 36. ratio of vertical change, to horizontal change, of a line

____ 37. a rate per one unit

____ 38. describes a line that slants downward from left to right

____ 39. difference in y-coordinates when computing slope

____ 40. ratio comparing amount of change in dependent variable with amount of change in independent variable in a

real-life situation.

7


a. income b. profit c. point of intersection d. break-even point

____ 41. location on a graph where two lines cross

____ 42. x-coordinate of intersection of the income equation and the production cost equation

____ 43. the amount of money a company earns

____ 44. amount of money that remains after the production costs are subtracted from income

Parallel and Perpendicular Lines in a Coordinate Plane

Match each definition or description to its corresponding term.

a. parallel linesb. perpendicular linesc. reciprocalsd. negative reciprocalse. horizontal line

f. vertical lineg. slopeh. point-slope formi. slope-intercept formj. y-intercept

____ 45. a line for which the slope is undefined

____ 46. two real numbers that have a product of 1

____ 47. y - y1 = m(x - x1 ) showing that a line passes through the point (x1 ,y1 ) and has slope m

____ 48. lines that intersect at a 90° angle

____ 49. y-coordinate of the point where a graph crosses the y-axis

____ 50. lines with the same slope and different y-intercepts

____ 51. y = mx + b showing that a line has slope m and y-intercept b

____ 52. two real numbers that have a product of 1

____ 53. ratio of vertical change to horizontal change

____ 54. a line described by y = b, where b is the y-intercept

8

Graphs and Solutions of Linear Systems

system of linear equations point of intersection solution linear system

slope reciprocals perpendicular parallel

Write the term that best completes each statement.

55. Two lines in the same plane are _______________________ if they do not intersect.

56. _______________________ lines are two lines that intersect to form a right angle.

57. Two non-zero numbers are _______________________ if their product is one.

58. The _______________________ of a non-vertical line is the ratio of the vertical change to the horizontal change.

59. A(n) _______________________ is two or more linear equations in the same variables.

60. A(n) _______________________ of a linear system is an ordered pair (x,y) that is a solution of both equations in thesystem.

61. The _______________________ is the location on a graph where two lines or functions intersect indicating that thevalues at that point are the same.

62. The solution of a _______________________ can be a single point, no points at all, or an infinite number of points.

orV bwh = V Bh =

S A . . = S A. . =

orbh bw hw + +2 2 2 Ph B+ 2

orV r h = � 2

= S A. . = S A. . =

rh r+2 2 2� �

rh B+2 2�

or

= 1 3

1 2

orV r h =

=

1 3 1 3

� 2 S A. . = r B( ) +1 2

2�

=

slant height of base

a apothem S.A. surface area

b base A area h height B area of base w width C circumference d diameter V volume r radius P perimeter

−180

−180 2( )n n

2( )n

Algebra 1 End-of-Course and Geometry End-of-Course Assessments Reference Sheet

Sum of the measures of the interior angles of a polygon =

=Measure of an interior angle of a regular polygon

where: n represents the number of sides

Florida Department of Education

| G–5

x x

x

x x

x

Algebra 1 End-of-Course and Geometry End-of-Course Assessments Reference Sheet

y2 y1

x2 x1

x1 y1 x2 y2

Slope formula

− −(x2 x1)2 + (y2 y1)

2

Midpoint between two points

( (

x1

x1

y1

y1

Quadratic formula

- −

Trigonometric Ratios opposite

opposite

hypotenuse

hypotenuse adjacent

adjacent

sin A°

tan A°

cos A°A°

Distance between two points

Slope-intercept form of a linear equation

Point-slope form of a linear equation

Special Right Triangles

Florida Department of Education

Topic: Create and interpret tables, graphs, and models to represent, analyze, and solve problems related to linear equations,

including analysis of domain, range, and the difference between discrete and continuous data.

Use a table of values to graph the following equations. Show all your work:

1.

y = 2x + 1

x y0

1

2

3

4

2. y = 4x – 1

x y0

1

2

3

4

3. y =ଵ

ଶx + 2

x y0

2

4

6

8

work!

Show all your

9

Solve the following equations for y, then use a table of values to graph them:

1. 6x + 2y = 12

x y-2

-1

0

2

4

6

2. 6x – 2y = 12

x y-6

-4

-2

0

2

4

3. 3x + 5y = 15

x y0

1

2

3

4

5

10
Show all your
work!

11

Topic: Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem.

Follow the indicated steps to work with the following equations:

1. 6x + 2y = 12

*Step 1: substitute x with zero, and solve for y________________________________________________________________________________________________

*Step 2: plot and label the answer on the coordinategrid

*Step 3: substitute y with zero, and solve for x________________________________________________________________________________________________


*Step 5: draw a line joining both points

2. 6x – 2y = 12






3. 3x + 5y = 15






Topic: Identify the solution to a system of linear equations using graphs.

Identify and label the solution of each system of linear equations:

1. 2.

3.

5.

12

4.

6.

(4, -3)

A product cost equation and income equation are shown on each graph. Estimate the break-even point. Then, use thespace below each graph to describe when the company will start to make a profit.

The break-even is the point whereexpenses and revenue intersect.

The graph shows that the production cost is $10 plus $2 per unit. The company sells their product for $3 each. Use thegraph to answer the questions. HINT: Use the definition of “break-even” and the graph to guide you on how to answerthese questions.

1. What is the production cost equation?

2. What is the income equation?

3. What is the break-even point?

4. What happens if the company sells fewer than 10 units?

5. What happens when the company sells more than 10units?

6. What does the ordered pair (10, 30) show?

Remember to use the

break-even illustration to

answer the questions!
13

14

Topic: Solve literal equations for a specified variable.

Use the inverse order of operations and solve for the indicated variable in the parenthesis.

1) P = IRT solve for (T) 2) A = 2(L + W) solve for (W)

3) y = 5x - 6 solve for (x) 4) 2x - 3y = 8 solve for (y)

5) x + y = 5 solve for (x) 6) y = mx + b solve for (b)

3

7) ax + by = c solve for (y) 8) A = 1/2h(b + c) solve for (b)

9) V = LWH solve for (L) 10) A = 4πr2 solve for (r2)

11) V = πr2h solve for (h) 12) 7x - y = 14 solve for (x)

13) A = x + y solve for (y) 14) R = E solve for (I)

2 I

15) x = yz solve for (z) 16) A = r solve for (L)

6 2L

17) A = a + b + c solve for (b) 18) 12x – 4y = 20 solve for (y)3

Show your

work!

15

Topic: Solve and graph one- and two-step inequalities in one variable.

When solving an inequality, use the inverse order of operations, as you would with an equation. However, remember that you mustuse the indicated inequality symbol.

When solving, the only different step is done when multiplying or dividing by a negative number- the inequality symbol must bereversed!

The solution to a linear inequality is not a single value- it is an interval of values that will make the inequality true. That is why whengraphing an inequality on a number line a closed or open dot is placed showing the beginning if the solution, and an arrow pointsat the side where all the other numbers that are part of the solution are located.

< > < >

○ ○ • •

Examples:

ݔ3 ≤ −3

ݔ3

3 ≤

−3

3≥ ݔ −1

2x – 5 > 9

2x – 5 > 9+ 5 + 52x > 142 2x > 7

Solve and graph the following inequalities

6x < 12 5x + 6 < 3

-3x - 4 ≥ 23 7x + 18 < 7

Use an open dot for these symbols

-3x + 2 > 8

-3x + 2 > 8-2 -2

-3x > 6-3 -3x < -2

1

4

Use a closed dot for these symbols

Remember

to reverse

the symbol!

16

Topic: Simplify real number expressions using the laws of exponents.

Write the term that best completes each statement

1. The __________________of a power is the number of times that the factor is repeatedly multiplied.2. An expression used to represent a factor as repeated multiplication is called a ________________.3. The __________________of a power is the repeated factor in a power

Word Bank

Base Power Exponent

Use the illustration to guide yourself in stating the base and the exponent of each power:

94

Base: ___

Exponent: ___

186

Base: ___

Exponent: ___

m5

Base: ___

Exponent: ___

27

Base: ___

Exponent: ___

൬1

2൰ଷ

Base: ___

Exponent: ___

-119

Base: ___

Exponent: ___

Consider the power following powers and their expanded notations:

43 = (4)(4)(4) = 64 -83 = (-8)(-8)(-8) = -512

Use the examples above and write each power in expanded notation, then evaluate to calculate the product.

(-4)5 54 -(83)

24 (-3)4 63

17

Topic: Perform operations on real numbers (including integer exponents, radicals, percents, scientific notation, absolute value, rational

numbers, and irrational numbers) using multi-step and real world problems.


(4)2(-5)3 = (4)(4)(-5)(-5)(-5) = -2000

Write each expression in expanded notation and evaluate to calculate the product.

(-4)4(7)2 (2)5(-6)2 -(6)3(4)2

-(4)3(8)2 -(5)3(9)2 (5)2(-3)3


(4)2(-5)3 = (4)(4)(-5)(-5)(-5) = -2000

Write each expression in expanded notation and evaluate to calculate the product.

(-4)4(7)2 (2)5(-6)2 -(6)3(4)2

-(4)3(8)2 -(5)3(9)2 (5)2(-3)3

Consider the following expressions:

(7)(7)(7)(7) = 74

y • y • y • y • y = y5

Use the examples above and write each expression using exponents.

(-2)(-2)(-2) = (5)(5)(5)(5)(5) = (6)(6) =

m • m • m = x • x • x • x • x • x = a • a •a • a =

Consider the following expression:

x2y2x3y5

x2 x3 y2y5

x2+3 y2+5 = x5 y7

Simplify each expression. Show your work.

1. a3b2ab4 2. 3mn3 • 8m6n7 3. (-m)2 • n3 • (-m)4 • n2

4. 3x3 • 2x2 • y3 • y (-b) • 4a • (-b)5 • 2a3 5. r2s6r3s2

work!

Show all your

18

Rules of exponents

Exponents have a few simple rules that can be used for simplifying expressions when the bases are the same.

Rule 1: Product rule of exponents

(x3)(x4) = (xxx)(xxxx) or (x3)(x4)

= xxxxxxx =x3 + 4

= x7 =x7

(-a)3(-a)4 34

(-c)(-c)3 n4

Rule 2: Power of a power

(x2)4 = (x2)(x2)(x2)(x2) or (x2)4

= (xx)(xx)(xx)(xx) = x(2)(4)

= xxxxxxxx = x8

= x8

1. (43)5 2.

4. -(65)2 5.

work!

• 35 (-2)2(-2)5

• n5 (-4)3(-4)7

(72)4

(22)3

When multiplying powers with the same base, just add the exponents

When applying an exponent to a power, keep the base

3. -(22)4

6. -(33)2

and multiply the exponents.

Show all your

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Rule 3: Quotient rule of exponents

= ି =

1. ళ

ఱ = 2.య

మ=

4.మ

మ= 5.

య

భ=

Rule 4: Power of a Quotient Property

ቆ

ቇ

= •

• =

ૡ

1. ቀ ళ

ఱቁଶ

= 2. ቀయ

మቁ

4. ቀమ

మቁହ

= 5. ቀయ

భቁ

When dividing powers with the same base, keep the base

3.௬ఴ

௬ర=

6.వ

ల=

ସ

=

ଷ

=

and subtract the exponents.

When applying an exponent to a quotient, multiply the

exponent by the exponents of the numerator and the

3. ቀ௬ఴ

௬రቁଷ

=

6. ቀవ

లቁଶ

=

denominator.

Show all your

work!

20

Entering Algebra 1 Summer Packet 2013

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