PreCalculus – Summer Packet

Entering into PreCalculus means entering into your first year of college preparatory mathematics. There is a major shift in expectation for students to be able to recall many skills at a moment’s notice. Certain concepts that have been taught to you over the previous years are assumed to be mastered. If you do not have these skills, you will find that you will consistently get problems incorrect next year as you make mistakes. It is frustrating for students who spend much of their homework time relearning algebra concepts in addition to learning how to tie the concepts together. This summer packet is intended for you to brush up and possibly relearn these topics.

On the following pages, you have assorted problems related to specific topics. Each problem should be done in the space provided. Rather than give you a textbook to remind you of the formulas and techniques necessary to solve the problem, there are a few websites listed that have full instructions on the techniques. If and when you are unsure of how to attempt these problems, use these websites. Don’t fake your way through these problems. You are only setting yourself up for a future struggle.

Realize also that many concepts are interrelated. This will be the focus throughout next year as we examine the mathematical relationships between topics numerically, algebraically and graphically. While you may be strong in one of these approaches, you must learn to view each topic from the other two approaches as well to achieve full understanding. Success on your tests and quizzes throughout the year will depend on you being able to do so.

You need to get off to a good start so spend some quality time on this packet this summer. Tear off this first page and return the remaining sheets stapled together. Be sure your name appears on the first sheet. Work needs to be shown when needed. Also, do not rely on the calculator to work through the majority of these problems. You will be tested without the calculator, so practice without the calculator.

This packet is to be completed by the first day back in school in the fall. You will be tested over this material on the second day of class. You will also be expected to efficiently work through the problems under a time constraint. Many students are not prepared for this expectation and find they do not have the time to check their answers like they are used to. Prepare accordingly.

It is a mistake to decide to do this now. Let it go until mid-summer. We want these techniques to be relatively fresh in your mind in the fall. But, do not wait to do them at the very last minute. These take time. If you do a few concepts a day, the whole packet will take you about a week to complete.

We hope you take this seriously as we sincerely wish for you to be successful throughout this next year. Your preparation over the summer will be rewarded in unexpected ways during the year.

Good Luck!

Note: Should you have any questions regarding this packet over the summer, Mrs Meldrum, Mr Barry, or Mrs Snipes will be available to answer your question via email. When you ask your question, please let them know what steps you have taken on your own so they may give you better guidance to solve that problem. Our email addresses are meldrumc@lisd.net, barryb@lisd.net, and GreenAbriana@lisd.net.

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

5

2

3. 32 )3( yx

4. 2

4

5

x

y 5.

2

1

xy

yx 6.

52

9

42

7

2

3

x

y

y

xy

2: Fractional Exponents Evaluate the following without a calculator

1. 3

2

8 2. 2

1

4

3. 24 16

4. 3 21000 5. 43 27

6. )25( 2

3

3: Simplifying Radicals Simplify and rationalize the following.

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

4. 8

3 5.

75

4 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 b

a 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

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Factoring Trinomials (a > 1) Factor each completely.

1) 3p2 − 2 p − 5

2)

2

n2 + 3

n − 9

3)

3

n2 − 8

n + 4 4)

5

n2 + 19

n + 12

5)

2

v2 + 11

v + 5 6)

2

n2 + 5

n + 2

7)

7

a2 + 53

a + 28 8)

9

k2 + 66

k + 21

-1-

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9)

15

n2 − 27

n − 6 10)

5

x2 − 18

x + 9

11)

4

n2 − 15

n − 25 12)

4

x2 − 35

x + 49

13)

4

n2 − 17

n + 4 14)

6

x2 + 7

x − 49

15)

6

x2 + 37

x + 6 16)

−6

a2 − 25

a − 25

17)

6

n2 + 5

n − 6 18)

16

b2 + 60

b − 100

-2-

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 1

5 2

x x 2.

5 2 1

2 6 3 12

x 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. 3

43 5 19x

12: Solving Rational Equations Solve the following for x

1. 3 9

62 2

xx 2.

2 2 8

3 3 6x x

3.

2

2 2

1 1 1

x

x x x

13: Solving Logarithmic Equations Solve the following for x

1. 3log 3 7x 2. 9

1log

2x 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. Find1

3f

Given 1 7

( )3 4

f x x , answer the following questions.

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

( )8

f x 6. Find 9

8f

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

_____1. 2

( ) 4 5 33

f x x A. Linear Function

_____2. 32( ) 4 5 3

3f x x B. Quadratic Function

_____3. 2 1

( ) 33 4 5

f xx

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) 33

f x x F. Square Root Function

_____7. 22( ) (4 5) 3

3f x x G. Rational Function

_____8. 2

( ) 4 5 33

f x x H. Polynomial Function

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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 xx

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. 2

7)(

xxf 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.

2

7,

2

3;

2

1,

2

33. 4,1;

2

3,

2

5

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

1. 92 xy 2. 3649 22 yx 3. 12

4 2

2

y

x

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

5

4,

3

2

(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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Name___________________________________

Period____Date________________

Trigonometric Ratios

Find the value of each trigonometric ratio.

1) tan Z

28

2135

Z Y

X

2) cos C

16

3034

C B

A

3) sin C

21

2835

C B

A

4) tan X

24

3240

X Y

Z

5) cos A

30

1634

AB

C

6) sin A

24

32

40

A

BC

7) sin Z

32

24

40

Z

Y X

8) sin C

4814

50C

B

A

9) cos Z

24

18

30

Z

Y X

10) tan C

36

27

45

C

B A

-1-

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Find the value of each trigonometric ratio to the nearest ten-thousandth.

11) cos Z

12 9

15Z

Y

X

12) cos C

2736

45 C

B

A

13) tan C

3040

50C

B

A

14) tan A

2021

29A

B

C

15) tan C

35

1237

CB

A

16) tan X

40 30

50X

Y

Z

17) sin Z

35

1237

ZY

X

18) sin Z

4030

50Z

Y

X

19) sin 48° 20) sin 38°

21) cos 61° 22) cos 51°

Critical thinking questions:

23) Can the sine of an angle ever equal 2?

Why or why not?24) sin x =

1

3

Find cos x.

-2-