PRE-CALCULUS for AP PREP
Summer Work and List of Topical Understandings
For students to successfully complete the objectives of the Pre-Calculus curriculum, the student
must demonstrate a high level of independence, capability, dedication, and effort. This summer
packet will help you maintain/improve your skills. This packet is a requirement for those
entering the Pre-Calculus course and is due on the first Friday of the semester. Your performance
(completeness and correctness) on this packet will count as your first homework grade for the
semester. Also, a pre-test on the material will be your first quiz grade of the Pre-Calculus course.
Complete as much of this packet on your own as you can, then get together with a friend, e-mail
your teacher, or “google” the topic. (math.com is also a good resource)SHOW YOUR BEST
WORK.
Requirements
The following are guidelines for completing the summer work packet…
 You must show all of your work. Use separate paper if necessary.
 Be sure all problems are neatly organized and all writing is legible.
 We expect you to come in with certain understandings that are prerequisite to Pre-Calculus.
A list of these topical understandings is below.
Topics within summer work…
 Exponent Rules
 Radical and Rationals
 Transformations of Functions
 Function Notation, Composite Functions and Inverse Functions
 All things Quadratics (solving, zeros, factoring, graphing, quadratic formula)
 Log Rules
 Triangle Trigonometry including right triangles, law of sines and law of cosines, evaluating
trig. functions and solving trig. equations
 Graphing Functions
 Interpreting and comprehending word problems
 Graphing, simplifying expressions, and solving equations of the following types:
trigonometric, rational, absolute value, logarithmic, exponential, polynomial/power, and
radical.
Finally, we suggest not waiting until the last two weeks of summer to begin on this packet. If
you spread it out, you will most likely retain the information much better. Once again this is
due, completed with quality, on the first day of class. It is intended to help you be successful in
the coming year. If you have any questions, we will check our e-mail throughout the summer, so
feel free to contact your teacher at vapniad@garnetvalleyschools.com or
romynk@garnetvalleyschools.com
GarnetValleyHigh School
Practice Set of Required Math Skills for
Pre-Calculus (AP Prep)
Skill 1: Exponent Rules
All students should be able to complete operations involving fractions and recall exponent facts
quickly and accurately without the use of a calculator.
Evaluate if x = 8, y = -2 and
z
2
1.
5.
3
 y
 
5
-34
1
2
2.
16 4
3.
(x – y)-2
4.
(xyz)0
6.
(-3)-4
7.
x4 = 16
8.
x3 = -343
Skill 2: Exponent Rules Con’t
All students should be able to apply the rules of exponents to simplify and expand expressions,
evaluate expressions and to solve equations. (Give answers with no negative exponents)
n 3
 49 
 
 4
y 4 2 x
y2

1
2
3.
 2x 
6.
x 5 y 3
w 0 z 1
9.
t 1  v
v
4 3
1.
x x
4.
y 12
y3
5.
7.
t 5 v 2
t 6 v 7
8.
x
2
x y
10.
x  4
11.
 x  5
12.
 2 x 
13.
6
 
5
14.

1
2 x 2
15.
3x 
3
2.
2
3 2
2
3
x9
3 2
Skill 3: Radicals and Rationals
All students should be able to simplify radicals and perform operations on radicals. (Give
answers with rationalized denominators)
Simplify, add or subtract.
1.
5.
300
x
2x

4 y 4 y
2.
192
2
3.
18
2
6.
4.
3 1

2 2
2
x

x  2x 4x  8
2
2 4

t v
7.
x
x3
x2
8.
5
3

x 4 x2
10.
6
1

x  2 x  15 x  3
1
1
x5
12.
3x 2  5 x  2
 x2  4  6 x  2
x
9.
2
x 1

y x
x2  y
xy
11.
2
Skill 4: Transformations of Functions.
Students should be able to identify, graph and write equations of parent functions based on
vertical and horizontal shifts, reflections over the x and y axis, and vertical and horizontal
stretches/compressions.
List the transformations of the parent function.
1.
f(x) = 2|x + 3| - 4
2.
g(x) =
2x
3.
h(x) = (-x – 1)2
Use the function f(x) shown below to graph the transformations in problems #4 - 8
y

4.
g(x) = f(x – 1)
5.
g(x) = f(x) + 1
6.
g(x) = ½f(x)
7.
g(x) = f(1/2x)
8.
g(x) = -½f(x + 2)



x












9.
Write the function that results from shifting f(x) = x 3 , 5 units to the right, 2 units down,
reflected over the x-axis, and a vertical stretch of 4.
Skill 5: Function Notation, Composite Functions and Inverse Functions.
Students should be able to comfortably evaluate functions, operations of functions, composition
of functions. Students should be able to find an inverse function and any necessary domain
restrictions so the inverse is itself a function.
Evaluate for the given functions.
1.
 2 x  4 x  2
f ( x)  
3x  2 x  2
a. f(1)
Given f ( x) 
b. f(-3)
2.
c. f(-2)
3
 x  1 if
g ( x)   2
 x  2 if

a. g(0)
x2
x2
b. g(2)
c. g(6)
5
, g(x) = x – 6 and h(x) = x2 – 4x – 12, find each function or value.
x3
3.
(f – g)(2)
4.
(h – g)(x)
5.
g
  8
h
6.
(gh)(5)
7.
g
8.
h(g(x))
f  (2)
Find the inverse function for each f(x). List any domain restrictions to f(x) so the inverse is a
function.
2
f ( x)  x  12
9.
12.
f ( x)  x 2  4
3
11.
f(x) = 3x
14.
f(x) = log2x
Skills 6: Quadratics.
Students should be able to factor quadratic and polynomial expressions, solve quadratic
equations, find the zeros and graph quadratics.
Factor:
b 2  11b  26
1.
5x2  7 x  6
4.
7.
9 + 8x – x2
2.
5.
8.
Solve:
10.
3x(x – 1) – x(x – 8) = 3
4 x 2  20
12.
x 4  81
100 x 2  75
x3  64
3.
6.
9.
11.
13.
36m 2  25n12
3x 2  5 x  2
4 x2  8x  4
2 x 2  7 x  15
x2 + 2x + 3 = 0
14.
16.
x 2  144  0
x3 – 9x2 – x + 9 = 0
15.
17.
x3  5 x  0
x 2  4 x  12
Graph the following. Give the coordinates of the vertex and the zeros.
18.
f(x) = -2(x – 3)2 + 4
19.
g(x) = ½x2 – 4x + 3
Graph the following.
20.
y > x2 + 2x + 1
Skill 7: Log Rules and Equations
All students should be able to apply the rules of logarithms to simplify and expand expressions,
evaluate expressions and to solve equations. (Give answers with no negative exponents)
1
ln e 5
log 2 16  log 2
log 10
1.
2.
3.
4
log 1
4.
5. log m x  log m 5  log m 10
6. log 6 x  log 6 2 x  1  2
Express as a single logarithm
7. log a 4  log a n
8. log x  log 5
Express in terms of log m and log n.
3
11. log m n
12. log n10m 
9. 5 log 2 w
10. ln v  2(ln 4  ln u )
Skill 8: Triangle Trigonometry
Students should be familiar with right triangle trigonometry, law of sines, law of cosines,
trigonometric functions, inverse trig. functions, special right triangles and the unit circle.
Find the exact value of the 6 trigonometric functions for each angle.
2

tan   
1.
2.
3.
  45
2
5
90° <  < 180°

4.
5.
6.    420
  210

4
7.
Find the exact vales of the 6 trig.
functions of  below.
10
8. Find the exact values of the 6 trig.
of an angle whose Terminal side
passes through  3, 8 .
6

Solve the following equations. Give all answers in the interval  0 , 360 
9.
sinx = ½
10.
tanx = 0
11.
2cos2x – 1 = 0
Evaluate. Give all answers in the interval 0, 2 

3
arcsin  

 2 
Solve the triangle (find all missing angles or sides)
15.
In ΔABC , b = 23.4, c = 14.7, C  37.2
13.
14.
arccos(1)
12.
arccot(-1)
16.
In ΔABC, a = 14 cm, b = 9 cm, c = 6 cm
17.
A lighthouse keeper observes that there is a 3 angle of depression to a sinking ship. If
the keeper is 19 m above the water, how far is the sinking ship from shore?
Find the missing sides of the 45-45-90 or 30-60-90 triangles.
45-45-90
1.
2.
3.
16 2
4.
5 2
16
6
30-60-90
5.
7.
6.
10 3
6
8.
9.
12
60
5 3
Skill 9: Graphing Functions
Students should be able to sketch graphs of non-linear functions, by using a table of values and
through transformations.
Graphing: Sketch a graph, showing the coordinates of two or more important points or
asymptotes.
1
y
1.
2.
3.
y  2x
y x
x
x
4.
ye
7.
y   x  3
8.
1
y 
3
y  2x 2
10.
y  x 2  4x  3
11.
13.
y  sin x
16.


y  2 cos x  
4

x
6.
y  x2  4
9.
y  2( x  4) 2  1
y  2 x4
12.
14.
y  cos x
15.
y  log 2 x
y = tanx
17.
y  3  sin 2 x
5.
2
Selected Answers:
Skill 1: #2) 8
#5) -81
Skill 2: #2) 2/7
#10) x2 + 8x + 16
x
x3
Skill 4: #1) Left 3, down 4, vertical stretch of 2
Skill 3: #4)
3
#14) 
x2
2
#9)
#2)
horizontal compression of 2
Skill 5: #1c) 8
#11) f 1 ( x)  log 3 x
#7) -1
Skill 6: #8) (x – 4)(x2 + 4x + 16)
Skill 7: #2) 5
#7) log a 4n
sin
Skill 8: #3)
cos
tan

2


2
2
1
0
 und

csc
sec

2
cot
2
#15) x = 0, + 5
1
 und

2
0
#9) x = 30 , 150