Avon High School
AP Calculus AB
Name ___________________________
Summer Review Packet
Period ______
1.) If f  x   4 x  x , find:
Topic A: Functions
2
a.) f  4  f  4
4 3
 r , find:
3
3
a.) V  
4
b.)
3
f 
2
c.)
f  x  h  f  x
2h
c.)
V  2r 
V r
2.) If V  r  
b.) V  r  1  V  r  1
3.) If f  x  and g  x  are given in the graph, find:
a.)
 f  g 3
b.) f  g  3 
x0
  x,
 2
0  x  2 , find:
4.) If f  x    x  1,

 x  2  2, x  2
a.) f  0  f  2
b.)
5  f  4 
c.) f  f  3 
Topic B: Domain and Range
Find the domain of the following functions using interval notation:
1.) f  x   3
x4
x 2  16
2.) y  x3  x 2  x
5.) f  x  
1
4x  4x  3
7.) f  t   t 3  1
8.) f  x  
5
10.) y  log  x  10
11.) y 
4.) y 
2
x2  x  2
2 x  14
x 2  49
3.) y 
x3  x 2  x
x
6.) y  2 x  9
9.) y  5x
12.) y 
2
4 x2
5 x
log x
Find the range of the following functions:
13.) y  x 4  x 2  1
14.) y  100x
15.) y  x 2  1  1
Find the domain and range of the following functions using interval notation:
16.)
17.)
18.)
Topic C: Graphs of Common 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. You may use a graphing calculator for some of them if you have access to one over
the summer. On the first day of class, you will be given a TI-Nspire to use (you could finish the last few then).
Another option is to find a graphing app or generate a table of values. Again, these are VERY important graphs
to know. Be very accurate with regards to “open circles” and “closed circles.”
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

 







Topic D: Even/Odd Functions and Symmetry
Show work to determine if the relation is even, odd, or neither.
1.) f  x   7
2.) f  x   2 x 2  4 x
4.) f  x  
7.) y  8 x 
x 1
1
8x
5.) f  x  
x2  1
8.) f  x   8x
3.) f  x   3x3  2 x
6.) f  x   8x
9.) f  x   8x  8x
Show work to determine if the graphs of these equations are symmetric to the x-axis, y-axis, or the origin.
10.) 4 x  1
11.) y 2  2 x 4  6
12.) 3x 2  4 y3
13.) x  y
14.) x  y
15.) x  y 2  2 y  1
Topic E: Function Transformations
If f  x   x 2  1 , describe in words what the following would do to the graph of f  x  :
1.) f  x   4
2.) f  x  4
3.)  f  x  2
4.) 5 f  x   3
5.) f  2 x 
6.) f  x 
Sketch the following graphs:
7.) y  2 f  x 
8.) y   f  x 
9.) y  f  x  1
10.) y  f  x   2
11.) y  f  x 
12.) y  f
Here is a graph of y  f  x  :
x
Topic F: Special Factorization
Factor completely.
1.) x3  8
2.) x3  8
3.) 27 x3  125 y 3
4.) x4  11x2  80
5.) ac  cd  ab  bd
6.) 2 x 2  50 y 2  20 xy
7.) x 2  12 x  36  9 y 2
8.) x3  xy 2  x 2 y  y 3
9.)  x  3  2 x  1   x  3   2 x  1
2
3
3
2
Topic G: Linear Functions
1.) Find the equation of the line in point-slope form, with the given slope, passing through the given point.
1
2 
1
a.) m  7,  3,  7 
b.) m   ,  2,  8 
c.) m  ,  6, 
2
3 
3
2.) Find the equation of the line in point-slope form, passing through the given points.
a.)  3, 6 ,  1, 2
b.)  7, 1 ,  3,  4
2 1 

c.)  2,  ,  , 1
3 2 

3.) Find the equations of the lines through the given point that are a.) parallel and b.) normal to the given line.
a.)  5,  3 , x  y  4
b.)  6, 2 , 5x  2 y  7
c.)  3,  4 , y  2
4.) Find the equation of the line in general form, containing the point  4,  2 and parallel to the line containing
the points  1, 4 and  2, 3 .
5.) Find k if the lines 3 x  5 y  9 and 2 x  ky  11 are a.) parallel and b.) perpendicular.
Topic H: Solving Quadratic Equations
1.) Solve each equation for x over the real number system.
1
0
4
a.) x2  7 x  18  0
b.) x 2  x 
d.) 12 x2  5x  2
e.) 20x2  56x  15  0
f.) 81x2  72 x  16  0
g.) x2  10 x  7
h.) 3x  4 x2  5
i.) 7 x2  7 x  2  0
k.) x3  5 x 2  5 x  25  0
l.) 2 x 4  15 x 3 18 x 2  0
j.) x 
1 17

x 4
c.) 2 x 2 72  0
2.) If y  x 2  kx  k , for what values of k will the quadratic have two real solutions?
Topic I: Asymptotes
For each function, find the equations of both the vertical asymptote(s) and horizontal asymptote (if it exists) and
the location of any holes.
1.) y 
x 1
x5
2.) y 
8
x2
3.) y 
2 x  16
x8
2 x2  6 x
4.) y  2
x  5x  6
x
5.) y  2
x  25
x2  5
6.) y  2
2 x  12
4  3x  x 2
7.) y 
3x 2
5x  1
8.) y  2
x  x 1
1  x  5x2
9.) y  2
x  x 1
x3
10.) y  2
x 4
x3  4 x
11.) y  3
x  2x2  4x  8
12.) y 
13.) y 
1
x

(Hint: express with a common denominator)
x x2
10 x  20
x  2x2  4x  8
3
Topic J: Negative and Fractional Exponents
Simplify and write with positive exponents.
2.)  12x5 
1.) 12 2 x 5
 4 
4.)  4 
x 
3
7.) 121x8 
 5x 3 
5.)  2 
 y 
1
10.)  x  y 
13.)
1
16 x
4
8.) 8x2 
2

3
4
 32 x 
2
2
9.)  32x 5 
3
 x  1
14.)
 x  1
2
1
6.)  x3  1
11.)  x3  3x 2  3x  1
2
3
2  4
3.)  4x 1 
2
1
1
2
2
2
3

1
3
12.) x x 2  x
5

2
15.)  x 2  22 
1
Topic K: Complex Fractions
Eliminate the complex fractions:
5
1.) 8
2

3
2
9
2.)
4
3
3
1
x
4.)
1
x
x
5.)
x
2
3.)
7 3

2 5
3
5
4
6.)
x 1  y 1
x y
4
1
x  x 1
7.)
x 2  x
1  x 1
1  x 2
1
3
 3x  4  4
8.) 3
3

4
2
2 x  2 x  1 2  2 x 2  2 x  1
9.)
 2 x  1
1
1
2
Topic L: Inverses
1.) Find the inverse of each of the following functions and show graphically that its inverse is a function.
a.) 2 x  6 y  1
b.) y  ax  b
c.) y  9  x2 , x  0
d.) y  1  x 3
e.) y 
9
x
f.) y 
2x  1
3  2x
2.) Find the inverse of each of the following functions and show that f  f 1  x    x
1
4
a.) f  x   x 
2
5
b.) f  x   x  4
2
x2
c.) f  x   2
x 1
3.) Without finding the inverse, find the domain and range of the inverse to f  x  
x 1
x2
Topic M: Adding Fractions and Solving Rational Equations
1.) Combine the following fractions:
a.)
2 1

3 x
b.)
1
1

x3 x3
c.)
5
5

2 x 3 x  15
d.)
2x 1
3x

x 1 2x  1
b.)
1
1
10

 2
x 3 x 3 x 9
d.)
2x 1
3x
x 2  11


x  1 2x  1 2x2  x  1
2.) Solve the equation for x.
a.)
2 1 5
 
3 x 6
c.)
5
5
5


2 x 3  x  5 x
Topic N: Absolute Value Equations
Solve the following equations:
1.) 4 x  8  20
2.) 1  7 x  13
3.) 8  2 x  2 x  40
4.) 4 x  5  5x  2  0
5.) x 2  2 x  1  7
6.) 12  x  x 2  12 x
Topic O: Solving Inequalities
Solve the following inequalities:
1.) 5  x  3  8  x  5
3.)
3
1
 x 1 
4
2
5.)  x  2  25
2
7.)
5
1

x6 x2
2.) 4 
5x
1

   2x  
3
2

4.) x  7  5  3x
6.) x3  4 x2
8.) Find the domain of:
x2  x  6
x4
Topic P: Exponential Functions and Logarithms
Simplify the following:
1.) log 2
1
4
4.) 5log5 40
7.) log 2
2
3
 log 2
3
32
1
2.) log8 4
3.) ln
5.) eln12
6.) log12 2  log12 9  log12 8
8.) log 1
3
4
 log 1 12
3
3
3
e2
 3
5
9.) log3
Solve the following:
10.) log5  3x  8  2


11.) log 9 x 2  x  3 
1
2
13.) log2  x  1  log2  x  3  5 14.) log5  x  3  log5 x  2
16.) 3x2  18
17.) e3 x1  10
12.) log  x  3  log5  2
15.) ln x 3  ln x 2 
18.) 8x  52 x1
1
2
Topic Q: Basic Right Angle Trigonometry
Solve the following:
If point P is on the terminal side of θ, find all 6 trigonometric functions of θ. (Answers need not be rationalized.)
1.) P  2,4
5
, θ in quadrant II,
13
find sin  and tan  .
3.) If cos   
2.) P

5, 2

2 10
, θ in quadrant III,
3
find sin  and cos .
4.) If cot  
5.) State the quadrant in which each of the following is true.
a.) sin   0 and cos  0
b.) csc  0 and cot   0
c.) tan   0 and sec  0
Topic R: Special Angles
Evaluate each of the following.
1.) sin 2 120  cos2 120
2.) 2tan 2 300  3sin 2 150  cos2 180
3.) cot 2 135  sin 2 210  5cos2 225
4.) cot  30  3tan 600  csc  450
2
3 

5.)  cos
 tan 
3
4 

6.)  sin
2


11
5  11
5 
 tan
 tan
 sin

6
6 
6
6 
Determine whether each of the following statements is true or false.
5
5
cos
1
cos






3
3

7.) sin  sin  sin   
8.)
5

5

6
3
6 3
tan 2
sec
1
3
3
3 
3
 3
 sin
9.) 2 
1  cos
2 
2
 2

0

cos3
10.)
4
4
 sin
3
3 0
2 4
cos
3
Topic S: Trigonometric Identities
Verify the following identities:
1.) 1  sin x 1  sin x   cos 2 x
3.)
1  sec x
  sec x
1  cos x
cos x  cos y sin x  sin y

0
5.)
sin x  sin y cos x  cos y
7.) csc  2 x  
csc x
2cos x
2.) sec2 x  3  tan 2 x  4
4.)
1
1

1
1  tan x 1  cot x
sin 3 x  cos3 x
6.)
 1  sin x cos x
sin x  cos x
8.)
cos  3x 
 1  4sin 2 x
cos x
Topic T: Solving Trigonometric Equations
Solve each equation on the interval  0, 2  .
1.) sin 2 x  sin x
2.) 3tan3 x  tan x
3.) sin 2 x  3cos2 x
4.) cos x  sin x tan x  2
5.) sin x  cos x
6.) 2cos2 x  sin x  1  0