Name______________________________________________Date___________Period________

Great Mills High School Math Department Summer Review

Algebra 2 CM, Algebra 2 Honors, STEM Algebra 2

This packet contains review material. You have been taught and tested on this material in prior math

courses. However, math is a cumulative study; it builds upon itself. Recalling these prerequisite

mathematical concepts and processes is critical to your success in this class.

The purpose of this review packet is three-fold:

1. Review/refresh prerequisite skills

2. Reinforce the expectation that you are prepared to take this course

3. Provide a smooth transition into a higher-level math course

This material will be assessed as follows:

1. Completion of this packet is a process grade for the class.

2. This packet will be returned to you and each problem will be reviewed.

Following review, you will take a process-grade group quiz.

3. Lastly, there will be a product-grade individual quiz on this material.

For each topic addressed, there are examples, explanations, and/or references, followed by a short

set of practice problems. You are expected to complete this packet on your own. Write all of your

final answers on the answer sheet provided. Only answers written on the answer sheet will be

graded. Be sure to show your work in the packet. If you need additional space, use lined notebook

paper and staple it to this packet.

This packet is due on August 19, 2015, the first day of school.

Topics:

1. Fractions

2. Simplify Polynomial Expressions

3. Solve Equations

4. Rules for Exponents

5. Radicals

6. Slope/Rate of Change

7. Graphing Lines

8. Quadratics

9. Right Triangles

Name______________________________________________________Date_____Period_____

Answer Sheet: Write all of your final answers on this sheet. Show your work in your packet or on

additional paper as needed. Only answers written on this sheet will be graded.

Practice Set 1:

1. ____________

2. ____________

3. ____________

4. ____________

Practice Set 2:

1. ____________

2. ____________

3. ____________

Practice Set 3:

1. ____________

2. ____________

3. ____________

4. ____________

Practice Set 4:

1. ____________

2. ____________

3. ____________

4. ____________

Practice Set 5:

1. ____________

2. ____________

3. ____________

4. ____________

Practice Set 6:

1. ____________

2. ____________

3. ____________

4. ____________

Practice Set 7:

1. ____________

2. ____________

3. See Graph

4. See Graph

Practice Set 8:

1. ____________ 8. ____________

2. ____________ 9. ____________

3. ____________ 10. ____________

4. ____________ 11. ____________

5. ____________ 12. ____________

6. ____________ 13. ____________

7. ____________ 14. See graph.

Practice Set 9:

1. ____________

2. ____________

3. ____________

4.____________

1. Fractions

Multiplying fractions:

bd

ac

d

c

b

a

21

10

7

5

3

2

Dividing fractions:

bc

ad

c

d

b

a

d

c

b

a

8

7

2

7

4

1

7

2

4

1

Adding or subtracting fractions without a common denominator:

db

cbda

db

cb

db

da

b

b

d

c

b

a

d

d

d

c

b

a

12

17

12

89

12

8

12

9

4

4

3

2

4

3

3

3

3

2

4

3

Practice Set I: Perform the following operations. Write answers in the lowest terms.

1. y

xz

y

x

4

3

5

12 2.

c

b

b

a

5

7

5

3

3. yx

65 4.

c

a

b

a 64

Need more help? Check out this website.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut3_fractions.htm

2. Simplify Polynomial Expressions

Combine like terms. “Like” terms have the same variable to the same power.

Example: 8x2 + 10x3 – 5x2 + 3x3

8x2 – 5x2 + 10x3 + 3x3

3x2 + 13x3

Apply the Distributive Property:

Example: 5(4x – 6)

5 • 4x – 5 • 6

20x – 30

Combine Like Terms and Apply the Distributive Property

Example: 4(6x – 3y) + 5(2x + 7y)

4 • 6x – 4 • 3y + 5 • 2x + 5 • 7y

24x – 12y + 10x + 35y

34x + 23y

Practice Set 2. Simplify. Show all work.

1. 3(5a – b) + 4(2a – 2b) 2. -5(4x – 7) + 13 – 6x

3. 8(3x + 5y – 6z) – 2(2x + 4z).

Need more help? Check out this website.

http://www.khanacademy.org/math/algebra/polynomials/polynomial_basics/v/simply-a-polynomial

3. Solve Equations

Simplify both sides of the equation.

Use addition/subtraction to move variables to one side, constants to the other.

Use multiplication/division to solve for the variable.

Example:

4(x + 7) = 22 – 2x

4x + 28 = 22 – 2x distribute the 4

+2x +2x add 2x to both sides

6x + 28 = 22

- 28 -28 subtract 28 from both sides

6x = -6

x = -1 divide both sides by 6

Practice Set 3. Solve each equation. Show all work.

1. 45x – 720 + 15x = 60 2. 8(3x – 4) = 232

2. -131 = -5(3x – 8) + 6x 4. – 7x – 22= 18 + 3x

Need more help? Check out these websites.

http://www.purplemath.com/modules/solvelin.htm

http://www.purplemath.com/modules/solvelin3.htm

4. Rules for Exponents

Practice Set 4. Simplify each expression.

1. (-3m2n)4 2. (x2y4)(x3y5)

3. cba

ba34

96

6

12 4.

3

4

2

12

3

x

x

Need more help? Check out this website.

http://www.mathsisfun.com/algebra/exponent-laws.html

5. Radicals

Practice Set 5. Simplify each radical. Show your work.

1. 486 2. 500

3. 475 4. √36𝑚4

Need more help? Check out this website.

http://hotmath.com/help/gt/genericalg1/section_8_1.html

6. Slope/Rate of Change

The slope of a line describes its steepness, or how it angles away from the horizontal. The slope of a

line is a rate of change and can expressed as a relationship between two variables, such as miles per

gallon or cost per pound.

12

12

xx

yy

run

rise

changehorizontal

changevertical

xinchange

yinchangemslope

Find the slope of the line passing through points (3, -1) and (-2, 5)

5

6

32

)1(5

m

Practice Set 6.

Find the slope of the line passing through each pair of points. Show your work.

1. (7, -9) , (-1, 5) 2. (4, 0) , (-6, 6)

F ind the slope of the lines represented on the graphs below.

3. 4.

Need more help? Check out these websites.

http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/Rate.htm

http://www.purplemath.com/modules/slopyint.htm

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

x

y

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

x

y

7. Write Linear Equations/Graph Lines

You can find the equation of a line from two points, or from one point and the slope. Slope-intercept

form, y = mx +b, where m is the slope and b is the y-intercept, is one form of a linear equation that is

particularly easy to graph.

Example: Write the equation of a line with a slope of 3 and passing through the point (5, 7)

y = mx + b

7 = 3(5) + b

-8 = b Equation: y = 3x – 8

Example: Write the equation of a line that passes through (2, 9) and (-1, 3)

23

6

21

93

m

52

5

)2(29

xy

b

b

bmxy

Problem Set 7. Write an equation, in slope-intercept form, using the given information.

1. m = -⅓ (-3, 6) 2. (-4, -1) (4, 5)

Graph the lines.

3. y = (2/3)x - 4 4. y = -3x + 3

Need more help? Check out this website.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut14_lineargr.htm

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

8. Quadratics

Factoring

Factoring with a leading coefficient of 1: x2 + bx + c

Example: x2 - 4x – 21 Identify b and c: b = - 4 c = -21

Find factors of – 21 that add to – 4

Factors of -21: 7 and -3, -7 and 3, 21 and – 1, -21 and 1. >>>> -7 + 3 = - 4

Use -7 and 3 as factors x2 – 4x - 21 = (x – 7)(x + 3) check using by multiplying out

x2 – 7x + 3x – 21 = x2 – 4x – 21

Practice Set 8 a: Factor the following completely

Decomposition: Factoring with a leading coefficient a ≠ 1: ax2 + bx + c

Example: 2x2 + 11x + 9 Identify a, b and c: a = 2, b = 11, c = 9

Multiply a ∙ c = (2)(9) = 18

Find factors of ac that add to b >>> Factors of 18 that add to 11.

Factors of 18: 2 and 9, -2 and – 9, 1 and 18, -1 and -18 >>>>>> 2 + 9 = 11

Rewrite the original problem by replacing b with the two new factors.

2x2 + 11x + 9 = 2x2 + 2x + 9x + 9

Group the first two terms and the second two terms: ( 2x2 + 2x ) + (9x + 9) **Note: make sure to write a + in

between the groupings. If 9x were negative, the negative would stay with the 9x)**

Factor out a common term from each grouping. 2x( x + 1) + 9( x + 1)

Factor out the common binomial from each grouping. (x + 1) ( 2x + 9)

Answer: 2x2 + 11x + 9 = (x + 1)(2x + 9)

Check by multiplying out: 2x2 + 9x + 1x + 9 = 2x2 + 11x + 9

Practice Set 8 b: Factor each completely.

Solving Quadratics: The solutions of a quadratic equation are called “zeros” or “roots”

To solve a quadratic, set the equation equal to 0. You may need to add or subtract all terms to one side or the

other. Then factor. After you have factored, you can apply the Zero Product Property.

Zero Product Property: If A * B = 0, then A = 0 and/or B = 0

Example:

3x2 + 11x + 8 = 0

Use decomposition: 3x2 + 3x + 8x + 8 = 0 (3x2 + 3x) + ( 8x + 8) = 0 3x(x + 1) + 8(x + 1) = 0

(3x + 8)(x +1) = 0 This means that either (3x + 8) =0 or (x + 1) = 0

Set each binomial equal to zero and solve for x.

3x + 8 = 0 >> 3x = - 8 >>> x = - 8 / 3 x + 1 = 0 >>> x = - 1

Check by plugging your answers back into the equation: 3(-8/3)2 + 11(-8/3) = - 8

3(-1)2 + 11(-1) = - 8

Practice Set 8 c: Solve each quadratic equation.

Graphing Quadratics in Standard Form:

Example: x2 - 6x + 8 = y

Start by factoring and solving for the zeros. x2 – 6x + 8= (x – 2)(x – 4)

Therefore the zeros are x = 2 , and x = 4. Graph these two points on the x – axis.

Find the x-value of the vertex by finding the halfway point between the zeros.

Add the zeros together, then divide by 2. (2 + 4) / 2 = 3

Plug the x – value of the vertex into the equation to find the y-value.

(3)2 -6(3) + 8 = 9 – 18 + 8 = -1 Therefor the vertex is at (3, -1). Plot this point.

The constant on the end of the function, c, is the y-intercept.

In this example, x2 - 6x + 8 = y, c = 8 Plot this point on the y-axis.

Connect all of the points with a smooth curve called a parabola.

Practice Set 8 d:

14) Sketch the graph of y = x2 + 6x + 8

Need more help? Check out this website.

www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-quadratic-expressions/v/factoring-quadratic-

expressions

www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-quadratic-

expressions/e/factoring_polynomials_2

www.khanacademy.org/math/algebra2/polynomial_and_rational/quad_factoring/v/example-1-solving-a-quadratic-equation-by-

factoring

www.khanacademy.org/math/algebra/quadratics/quadratics-square-root/v/solving-quadratic-equations-by-square-roots

www.khanacademy.org/math/algebra2/polynomial_and_rational/quad_formula_tutorial/v/using-the-quadratic-formula

9. Right Triangles

Example: Given that a = 6 and b = 8, find the length of the hypotenuse.

10,100100643686 2222222 cccsobac

Problem Set 8. Given right triangle ABC, where C = 90º, solve for the missing side. Show all work.

1. a = 12, b = 5, find c. 2. b = 15, c = 17, find a.

Given that a = 9, b = 40, and c = 41, find the trig ratio.

3. sin B 4. tan A

Need more help? Check out these websites.

http://www.mathsisfun.com/algebra/trig-finding-angle-right-triangle.html

http://www.purplemath.com/modules/basirati.htm

a

bB

c

bB

c

bB

b

aA

c

bA

c

aA

adj

opp

hyp

adj

hyp

opp

SOHCAHTOA

cbaTheoremnPythagorea

tancossin

tancossin

tancossin

: 222