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Avon High School
VU Calculus
Name ___________________________
Summer Review Packet
Period ______
Topic 1: Fractional & Negative Exponents
Simplify using only positive exponents.
1.) 3x 3
 2   2 
2.) 2 

2
 2  x    2  x  
 1
4
1 
3.)  2  1 1  2 
x y
y 
x
1
2
Topic 2: Domain
Find the domain of the following functions:
1.) y 
4.) y 
3x  2
4x  1
2x  9
2x  9
2.) y 
x2  5x  6
x 2  3x  18
5.) y  log  2 x  12 
3.) y 
6.) y 
x 3  x 3
x
,
cos x
0  x  2
Topic 3: Solving Inequalities: Absolute Value
Write the following absolute value equations as piecewise equations:
1.) y  2 x  4
2.) y  6  2 x  1
3.) y  4 x  1  2 x  3
Solve the following absolute value inequalities:
4.) x  3  12
5.) x  3  4
6.) 3x  4  2
Topic 4: Solving Inequalities: Quadratic
Write the following absolute value equations as piecewise equations:
1.) y  x 2  1
2.) y  x 2  x  12
3.) y  x 2  4 x  4
Solve the following by factoring and making appropriate sign charts:
4.) x2  16  0
5.) x2  6 x  16  0
6.) 2 x2  5x  3
Topic 5: Special Factoring Formulas
Factor completely.
1.) x3  8
2.) x4  11x2  80
3.) x 3  xy 2  x 2 y  y 3
Topic 6: Function Transformation
Here is a graph of y  f  x  :
Sketch the following graphs:
1.) y  2 f  x 
2.) y   f  x 
4.) y  f  x  2
5.) y  f  x 
3.) y  f  x  1
6.) y  f
x
Topic 7: Factor Theorem (p over q method)
Use the p over q method and synthetic division to factor the polynomial P  x  . Then solve P  x   0 .
1.) P  x   x3  4 x 2  x  6
2.) P  x   2 x3  3x 2  18 x  8
Topic 8: Even/Odd Functions
Show work to determine if the relation is even, odd, or neither.
1.) f  x   2 x 2  7
2.) f  x   4 x3  2 x
3.) f  x   4 x 2  4 x  4
Topic 9: Solving Quadratic Functions and Quadratic Formula
Solve each equation over the real number system.
1.) 7 x2  3x  0
2.) x 2 6 x  4  0
3.) x 
1 13

x 6
Topic 10: Asymptotes
For each function, find the equations of both the vertical asymptote(s) and horizontal asymptote (if it exists).
1.) y 
x
x3
2.) y 
x4
x2  1
3.) y 
x4
x2  1
Topic 11: Complex Fractions
Simplify the following:
1.)
x
x
1
2
1
4
x
2.)
1
2
x
3

x
3.)
4

x
4
y
3
y
Topic 12: Composition of Functions
If f  x   x 2 and g  x   2 x  1, find the following:
1.) f  g  2  
2.) g  f  2  
3.) f  g  x  
Topic 13: Solving Rational Equations
Solve each equation for x.
1.)
2 5 1
 
3 6 x
2.)
x 1 x 1

1
3
2
3.)
2
1
16

 2
x  5 x  5 x  25
Topic 14: Basic Right Angle Trigonometry
Solve the following:
If point P is on the terminal side of θ, find all 6 trigonometric functions of θ. Draw a picture.
1.) P  2,4
2.) P
5
, θ in quadrant II,
13
find sin  and tan  .

5, 2

4.) If cot   3 , θ in quadrant III,
3.) If cos   
find sin  and cos .
Find the exact value of the following without a calculator:
5.) sin 2 225  cos2 300
6.)  6sec180  4cot 90 
2
7.)  4cos30  6sin120
Topic 15: Solving Trigonometric Equations
Solve each equation on the interval  0, 2  .
1.) sin x 
1
2
2.) cos2 x  cos x
3.) 4sin 2 x  1
2
Topic 16: Sketches of Basic Functions
Sketch each of the following as accurately as possible. You will need to be VERY familiar with each of these
graphs throughout the year.
1. y  x
2. y  x 2
3. y  x3
4. y  x
5. y  x
6. y 
x
x
7. y  x
1
8. y  x
3
9. y  sin x
2
3
10. y  cos x

 







 








 











13. y  sec x
14. y  csc x





 

12. y  cot x




11. y  tan x
 








 



15. y  e x
16. y  ln x
17. y 
1
x
18. y  x
19. y 
1
x2
20. y  2 x
21. y  4  x 2
22. y 
sin x
x

 






