PreCalculus Readiness Summer

Packet El Segundo High School

This packet is designed for those who have completed Algebra 2

and will be enrolled in PreCalculus in the upcoming fall semester.

Use www.google.com to search for skill-related videos on any of these topics. Use www.khanacademy.org to find specific math related topics with accompanying videos. Use www.wolframalpha.com to see step by step answers for specific questions you may have. Topics Exponents 1. Basic Rules 2. Fractional Exponents 3. Simplifying Radicals Factoring 4. Factoring by GCF 5. Factoring quadratic expressions 6. Special Factoring formulas 7. Factoring through synthetic division Equations/Inequalities 8. Solving linear equations. 9. Solving quadratic equations by factoring 10. Solving quadratic equations by quadratic formula 11. Solving radical equations 12. Solving rational equations 13. Solving logarithmic equations Functions 14. Function notation 15. Function names 16. Function Operations. Graphing 17. Function transformations 18. Graphing Parent Functions by T-chart  using  “smart”  points 19. Basic Graphing using “smart”  points General Topics 20. Distance and Midpoint formulas 21. Intercepts 22. Equations of lines 23. Pythagorean Theorem 24. Common Algebraic Errors

Name __________________________________ 1: Exponent Rules Simplify the following

1. 32 )2(� 2. 2

52 �

¸¹·

¨©§� 3. 32 )3( �yx

4. 2

4

5 �

�

xy 5. 2

1

�

�

xyyx 6. 52

9

427

23

xy

yxy �

��

2: Fractional Exponents Evaluate the following without a calculator

1. 32

8 2. 21

4�

3. � �24 16

4. 3 21000 5. � � 43 27�

� 6. )25( 23

��

3: Simplifying Radicals Simplify and rationalize the following.

1. 80 2. 4 32 3. 3 354x

4. 83

5. 754

6. 2134 �

4: Factoring by GCF Factor the following completely 1. 24 93 xx � 2. yxxy 142849 �� 3. 2453 1218 yxyx � 5: Factoring Quadratic Expressions Factor the following completely 1. 232 �� xx 2. 652 �� xx 3. 352 2 �� xx 4. 483 2 �� xx 5. 10173 2 �� xx 6. 61910 2 �� xx 6: Special Factoring

2 2 2 3 3 2 2

2 2 2 3 3 2 2

2 2

2 ( ) ( )( )

2 ( ) ( )( )

( )( )

a ab b a b a b a b a ab ba ab b a b a b a b a ab b

a b a b a b

� � � � � � �

� � � � � � �

� � �

Factor the following completely 1. 25204 2 �� xx 2. 22 94249 yxyx �� 3. 8116 4 �x 4. 3 8x � 5. 3 3125x y� 6. 664 27y�

7: Factoring through Synthetic Division Use synthetic division to factor as indicated. 1. ))(1(124 23 � ��� xxxx 2. ))(1(752 3 � �� xxx 3. ))(2(23 234 � ���� xxxxx 4. ))(12(134 24 � �� xxx 8: Solving Linear Equations Solve the following for the unknown variable.

1. 2 1 3 15 2x x� �

2. 5 2 12 6 3 12x x x� � 3. 3( 8) 4 5 ( 7)x x x x� � � �

9: Solving Quadratic Equations by Factoring Factor to solve for x. 1. 2 5 6 0x x� � 2. 28 6 5 0x x� � 3. 211 14 16 0x x� �

10: Solving Quadratic Equations using the Quadratic Formula For each equation, solve for the indicated expression. 1. 22 4 1 0x x for x� � 2. 22 2 3 0x x for x� � 3. 4 2 24 2 0x x for x� � 11: Solving Radical Equations Solve the following for x.

1. 3 1x x � 2. 3 2 1 7x � 3. 343 5 19x �

12: Solving Rational Equations Solve the following for x

1. 3 96

2 2x

x� 2. 2 2 8

3 3 6x x�

� 3. 2

2 21 1 1

xx x x

� � � �

13: Solving Logarithmic Equations Solve the following for x

1. 3log 3 7x 2. 91

log2

x 3. 32log ( 1) 4x�

14: Function Notation Given 2( )f x x x � � , answer the following questions.

1. Find (0)f 2. Find ( ) 0f x 3. Find 13

f § ·�¨ ¸© ¹

Given 1 7

( )3 4

f x x � , answer the following questions.

4. Find the zeros of ( )f x 5. Solve 1( )

8f x 6. Find 9

8f § ·�¨ ¸© ¹

15: Function Names Match the following equations to their description.

_____1. 2( ) 4 5 3

3f x x � � A. Linear Function

_____2. 32( ) 4 5 33

f x x � � B. Quadratic Function

_____3. 2 1( ) 3

3 4 5f x

x � �

� C. Absolute Value Function

_____4. 4 22( ) (4 5) 3(4 5) 2

3f x x x � � � � D. Cubic Function

_____5. 32( ) (4 5) 3

3f x x � � E. Cube Root Function

_____6. 2( ) (4 5) 3

3f x x � � F. Square Root Function

_____7. 22( ) (4 5) 3

3f x x � � G. Rational Function

_____8. 2( ) 4 5 3

3f x x � � H. Polynomial Function

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

16: Function Operations Perform the following function operations if 2( ) 2f x x and ( ) 3 4g x x � 1. � �( )f g x 2. � �( )g f x 3. ( )( )f g x� 4. � �( )f f x 5. � �( )g g x 6. Find � �( ) 0g g x 17: Function Transformation Use the graph of ( )y f x at the right to sketch the following transformations. 1. 2 ( )y f x 2. ( )y f x � 3. ( 1)y f x � 4. ( ) 2y f x � 5. ( )y f x � 6. 2 ( 2) 1y f x � � �

18: Graphing Parent Functions using T-Charts Graph the following using a T-Chart  with  “smart”  values.    State  the  Domain  and  Range  of  each  function. 1. 2( ) xf x 2. ( )f x x 3. ( )f x x D: R: D: R: D: R:

4. 3( ) xf x 5. 3( )f x x 6. 1( )f x

x

D: R: D: R: D: R: 7. ( ) xf x 8. ( ) 2xf x 9. 2( ) logf x x D: R: D: R: D: R:

19: Basic Graphing Choosing  “Smart”  Points Fill in the T-chart using at least 3 smart x-values (that enable you to find exact points)

1. xxf � 3)( 2. 27

)(�

x

xf 3. 43)(x

xf

20: Distance and Midpoint Formulas Find the distance between the two points. Then find the midpoint between the two points.

1. )1,6();5,2( �� 2. ¸¹·

¨©§�¸

¹·

¨©§ �

27,23

;21

,23

3. � �4,1;23

,25

�¸¹·

¨©§ �

21: Intercepts Use the following equations to find the x and y intercept(s)

1. 92 � xy 2. 3649 22 � yx 3. 124 2

2

�¸¹·

¨©§ � yx

22: Equations of Lines Find the equation of the line that has the given characteristics. Leave your answer in the form indicated.

1. 3 2; int :4 3

slope y � � 2. Parallel to 432 � yx through 3. Perpendicular to 2374 � yx

(Standard Form) � �6,3� through ¸¹·

¨©§ �

54

,32

(Slope-intercept form) (Point-Slope Form)

c

a

b

23: Pythagorean Theorem Use the diagram at the right to answer the following questions. Be sure to simplify. 1. Find b if a = 54 , c = 2 2. Find c if a = 32 , b=6 3. If a = c, and b = 10, find a 24: Algebraic Errors to Avoid

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Worksheet by Kuta Software LLC

Precalculus

Assignment

Name___________________________________ ID: 1

Date________________ Period____

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Plot each point in the complex plane using rectangular coordinates.

1) -5 - 3i 2) -5 + 6i

3) 5 - 3i 4) 1 - 2i

Simplify. Write your answer in rectangular form.

5) (2 - 4i)(-2 + 4i) 6) (4 - 3i)(-5 + 3i)

7) (-1 - 2i) × i 8) (-3 - 6i)(3 + i)

9) 2 + 6i

6 - 4i10)

-4 + 5i

-4 - 4i

11) 1 - 6i

2 + 4i12)

-4 - 4i

1 + 4i

Solutions 1: Exponent Rules 1.) 64 2.) !!"! 3.) !

!"!!!! 4.) !!

!!! 5.) !!

!! 6.) !!"

!!!! 2: Fractional Exponents 1.) 4 2.) ½ 3.) 4 4.) 100 5.) !!" 6.) !!!"# 3: Simplifying Radicals 1.) 4 5 2.) 2 2! 3.) 3x 2! 4.) ! !

! 5.) ! !!" 6.) 12 7

4: Factoring by GCF 1.) 3x2(x2 – 3) 2.) 7(7xy + 4x – 2y) 3.) 6x3y2(3y3 – 2x) 5: Factoring Quadratic Expressions 1.) (x – 2)(x – 1) 2.) (x + 6)(x – 1) 3.) (2x – 1)(x + 3) 4.) (3x – 2)(x – 2) 5.) (3x + 2)(x + 5) 6.) (5x – 2)(2x – 3) 6: Special Factoring 1.) (2x – 5)2 2.) (7x + 3y)2 3.) (4x2 + 9)(2x + 3)(2x – 3) 4.) (x – 2)(x2 + 2x + 4) 5.) (5x + y)(25x2 – 5xy + y2) 6.) (4 – 3y2)(16 + 12y2 + 9y4) 7: Factoring through Synthetic Division 1.) x2 – 3x – 1 2.) 2x2 – 2x + 7 3.) x3 – x2 – x – 1 4.) 2(x2 + 1)(2x + 1) 8: Solving Linear Equations 1.) !!!! 2.) !! 3.) !"! 9: Solving Quadratic Equations by Factoring 1.) -3, -2 2.) !!, -½ 3.) !!!! , 2 10: Solving Quadratic Equations using the Quadratic Formula

1.) !!±! !! ! 2.) !!!±! !! 3.) 2 ±! 2

11: Solving Radical Equations

1.) !!±! !"!" 2.) 2 3.) 16 12: Solving Rational Equations 1.) ¼, -1 2.) 2, 3 3.) -4 13: Solving Logarithmic Equations 1.) 7 2.) 3 3.) 8

14: Function Notation 1.) 0 2.) 0, 1 3.) !!!

4.) !!"! 5.) !!"! 6.) !!! 15: Function Names 1.) C 2.) E 3.) G 4.) H 5.) D 6.) A 7.) B 8.) F 16: Function Operations 1.) 32x2 – 48x + 18 2.) -8x2 + 3 3.) 2x2 + 4x – 3 4.) 8x4 5.) 16x – 9 6.) !!" 17: Function Transformations

18: Graphing Parent Functions 1.)

D: (-∞,∞) R: [0, ∞)

2.)

D: [0, ∞) R: [0, ∞)

3.)

D: (-∞,∞) R: [0, ∞)

4.)

D: (-∞,∞) R: (-∞,∞)

5.)

D: (-∞,∞) R: (-∞,∞)

6.)

D: x!≠ 0 R: y ≠ 0

7.)

D: (-∞,∞) R: (-∞,∞)

8.)

D: (-∞,∞) R: (0,∞)

9.)

D: (0,∞)

R: (-∞,∞) 19: Basic Graphing (Answers may vary) 1.)

x y 2 1 3 0 4 undefined

2.) x y 1 7 2 undefined 3 -7

3.) x y -1 (!!)

1/4

0 1 1 31/4

20: Distance and Midpoint Formulas

1.) d = 10; mp = (2, 2) 2.) d = 5; mp = (0, !!) 3.) d = !"! ; mp = (!!,

!!! )

21: Intercepts 1.) x-intercept: (-9, 0)

2.) x- intercepts: (3, 0), (-3, 0)

3.) x-intercepts: (-6, 0), (-2, 0)

y-intercepts: (0, 3), (0, -3) y- intercepts: (0, 2), (0, -2) y-intercept(s): none 22: Equations of Lines 1.) -9x + 12y = -8 2.) y = !!! x + 4 3.) y + !! = !!! (x – !!) 23: Pythagorean Theorem 1.) 2 21 2.) 2 6 3.) 5 2 24: Algebraic Errors to Avoid Please be sure you’ve read over this section carefully; great points here!

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Worksheet by Kuta Software LLC

Answers to Assignment (ID: 1)

1)

Real

Imaginary

-4 -2 2 4

-4

-2

2

4

2)

Real

Imaginary

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

3)

Real

Imaginary

-4 -2 2 4

-4

-2

2

4

4)

Real

Imaginary

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

5) 12 + 16i 6) -11 + 27i

7) 2 - i 8) -3 - 21i9) -

3

13 +

11

13i 10) -

1

8 -

9

8i

11) -11

10 -

4

5i 12) -

20

17 +

12

17i