Summer Review Packet – Pre AP Precalculus
Directions for this packet: In the first part of this packet, you will be required to solve for the indicated
variable in various situations. On a separate sheet of paper, show your work for each problem, and circle
your answer. This is not an option!
While you will most likely be able to solve a great number of these on your own, some may require
assistance from outside sources. You are strongly encouraged to search the internet, work with others, or
find other resources to solve these problems.
The remainder of the packet contains various other exercises requiring simplification or investigation of
the graphs of various functions shown with these exercises.
**If you wish, a calculator will be available at the high school. You will be required to turn in this
calculator in class on the first day of school.**
You are expected to finish as much of this as possible this summer. We will be spending a part of the first
week of school going over any questions you may have. This work will receive multiple daily grades and
a quiz grade.
GOOD LUCK!
Mrs. O’Rear
Are You Ready for
Pre AP Precalculus?
SOLVING
Solve each of the following for “x.” Remember that all work is to be done on separate paper.
SHOW YOUR WORK, and circle your answers.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
−4 𝑥 − 2 + 3 𝑥 + 1 = 7
−3 𝑥 − 2 = 6 4𝑥 + 1
15 𝑥 − 2 = −13 1 + 𝑥 + 11
15 4 − 𝑥 = 5 10 + 2𝑥
−2𝑥 = 2 𝑥 + 1 + 3
2 4𝑥 − 𝑥 − 1 = 2 1 − 𝑥
1 𝑥−5=1
2
6𝑥 = 2 3 5𝑥 − 2
!!!
!!
= !
!
4𝑥 − 1 = 11
Solve each for the indicated variable.
11.
12.
13.
14.
15.
16.
17.
18.
Solve for r: 𝐴 = 2𝜋𝑟ℎ
Solve for w: 𝑉 = 𝑙𝑤ℎ
Solve for w: 𝑃 = 2𝑙 + 2𝑤
Solve for t: 𝐴 = 𝑃 + 𝑃𝑟𝑡
!
!
Solve for b: ! = !
!
Solve for h: 𝑉 = ! 𝑟 ! ℎ
!
Solve for x: 𝑝 = 12 − !"""
Solve for r: 𝑆 = 4𝜋𝑟 !
FRACTION REVIEW
Simplify each of the following by:
a) Addition or subtraction. Remember the rules for adding or subtracting fractions. If you
don’t remember the rules, look them up!
19.
20.
21.
22.
23.
24.
25.
26.
!
!
+!
!
!
!
!
+4
!
− !"
!"
!
!
!
!
!
!
!
−!
−𝑥
!
+ !!
!
!
− !!!
!!!
!
!!! !
!
+ !!!
b) Multiplication or division. Again, remember the rules for multiplying and dividing
fractions. Look the rules up if you don’t remember them!
27.
28.
29.
30.
31.
32.
33.
34.
! !
∙
! !
!
!
∙4
!!!
!
!
∙ !!!
!!
!!!
∙
!!! !!!
!
!
÷4
!
!
÷!
!
! !
! !
!
! !
35.
36.
37.
38.
! !
!
! !
! !"
!!!
! !!!
!
!!!
!
!!
!!!
!!
c) Answer questions a. – c. regarding each of the following functions.
39.
!!!
𝑓 𝑥 = !!!
a. Is the point (3, 14) on the graph of f ?
b. If x = 4, what is f (x)? What point is on the graph of f ?
c. If f (x) = 2, what is x?
40.
!!
𝑓 𝑥 = ! ! !!
a. Is the point (-1, 1) on the graph of f ?
b. If x = 2, what is 𝑓 𝑥 ?
c. If 𝑓 𝑥 = 1, what is x?
41.
𝑓 𝑥 =
! ! !!
!!!
a. Is the point (1, 3/5) on the graph of f ?
b. If x = 0, what is 𝑓 𝑥 ?
!
c. If 𝑓 𝑥 = !, what is x?
42.
!!
𝑓 𝑥 = !!!
a. Is the point 1 2 , −2 3 on the graph of f ?
b. If x = 4, what is 𝑓 𝑥 ?
c. If 𝑓 𝑥 = 1, what is x?
QUADRATICS
Solve each of the following by factoring.
43.
𝑥 ! − 10𝑥 = 24
44.
4𝑥 ! − 9 = 0
45.
9𝑥 ! − 7𝑥 = 0
46.
30𝑥 ! − 2𝑥 = 4
Simplify each of the following. Remember: 𝑖 = −1, and 𝑖 ! = −1.
47.
6 + −49
!
48.
8𝑖
49.
5 + 7𝑖 + 4𝑖
50.
3 − 8𝑖 − 2 + 𝑖
51.
6 + 4𝑖 − 9 − 4𝑖
52.
2 − 9𝑖 + 4 + 9𝑖
53.
4 + 3𝑖 1 − 7𝑖
54.
9 + 5𝑖
55.
2 + 5𝑖 2 − 5𝑖
56.
57.
!
!!
!!!!
!!!!
!!!!
Solve by using the quadratic formula:
58.
2𝑥 ! + 5𝑥 − 7 = 0
59.
𝑥 ! + 6𝑥 − 5 = 0
60.
2𝑥 ! + 8𝑥 = −12
61.
3𝑥 ! = −4𝑥 − 10
Evaluate the discriminant of each equation. Determine how many solutions and whether they are
real or imaginary.
62.
𝑥 ! + 4𝑥 + 5 = 0
63.
2𝑥 ! + 7𝑥 − 15 = 0
64.
4𝑥 ! + 20𝑥 + 25 = 0
Write each quadratic equation in standard (vertex-intercept) form by completing the square.
Then graph.
65.
𝑦 = 𝑥 ! − 4𝑥 + 6
66.
𝑦 = −2𝑥 ! + 12𝑥 − 15
Write a quadratic equation for each pair of roots.
67.
-6 and -9
68.
-1 and 3/2
69.
4 (double root)
POLYNOMIALS
Simplify by using the indicated operations. Write each answer in descending order of exponents.
70.
−8𝑑 ! − 7 + −𝑑 ! − 6
71.
30𝑥 ! − 49𝑥 ! + 7𝑥 + 50𝑥 ! − 75 − 60𝑥 !
72.
5𝑥 ! − 6𝑥 + 8 − 3𝑥 ! − 9
73.
−3𝑥 ! + 7𝑥 ! − 8 − −5𝑥 ! + 9𝑥 ! − 8𝑥 + 19
74.
5𝑥 ! 6𝑥 − 2
75.
𝑥! + 1
!
76.
2𝑥 − 3 2𝑥 + 4 2𝑥 − 1
Factor each of the polynomials completely.
77.
𝑥 ! + 64
78.
𝑥 ! + 3𝑥 ! + 2𝑥
79.
3𝑥 ! − 12𝑥
80.
2𝑥 ! − 5𝑥 − 12
81.
𝑥 ! + 3𝑥 ! − 4𝑥 − 12
82.
125𝑥 ! − 27
83.
6𝑥 ! − 48𝑥
84.
3𝑥 ! − 5𝑥 ! + 6𝑥 − 15
Divide using long division.
85.
9𝑥 ! − 18𝑥 ! − 𝑥 + 2 ÷ 3𝑥 + 1
86.
3𝑥 ! − 5𝑥 + 1 ÷ 𝑥 + 9
Divide using synthetic division.
87.
6𝑥 ! − 4𝑥 ! + 14𝑥 − 8 ÷ 𝑥 + 2
88.
𝑥 ! + 3𝑥 ! + 𝑥 + 4 ÷ 𝑥 + 3
89.
3𝑥 ! − 5𝑥 ! + 2𝑥 ! + 3𝑥 − 2 ÷ 3𝑥 − 2
For each of the following: a) list all possible rational zeros; and b) find all actual rational roots.
90.
𝑥 ! + 4𝑥 ! + 𝑥 − 6 = 0
91.
2𝑥 ! − 5𝑥 ! − 17𝑥 ! + 41𝑥 − 21 = 0
LINEAR FUNCTIONS
Graph:
92.
𝑦 = −3𝑥 + 7
93.
𝑦 = −𝑥
94.
𝑦=5
95.
3𝑥 − 2𝑦 = 6
96.
𝑥 = −2
97.
𝑦 = 3 5𝑥 − 4
98.
2𝑥 + 5𝑦 = −5
99.
3𝑥 − 4𝑦 = 10
Find the slope of each line containing the following pairs of points.
100.
(3, -2) (-5, -6)
101.
(1, 7) (3, 4)
102.
(2, 5) (2, 9)
103.
(1, 8) (-3, 8)
104.
! !
! !
! !
! !
,
,
Determine whether the lines containing the following pairs of points are parallel, perpendicular,
or neither.
105.
Line 1: (4, 3) (1, 2)
Line 2: (2, 5) (4, 11)
106.
Line 1: (3, 7) (-1, 6)
Lines 2: (5, -5) (4, -9)
Find the equation of the line with the given slope and containing the given point.
107.
𝑚 = 1 2, (4, 5)
108.
𝑚 = −2 3, (1, 1)
Find the equation of the line containing the two given points.
109.
(4, 5) (7, 3)
110.
(-3, 2) (5, 2)
111.
(1, 9) (1, 4)
Find the midpoint of the segment with the given endpoints; then find the distance between the
two given points.
112.
(-4, 3) (2, 4)
113.
(5, -1) (9, 2)
The following is a graph that represents the distance d (in miles) that Kevin is from home as a
function of time t (in hours). Answer questions 114-122 based on the graph.
(2, 3) (2.5, 3) (3.9,2.8) (2.8, 0) (3, 0) 114.
How many times did Kevin return home?
Describe Kevin’s distance from home:
115.
from t = 0 to t = 2
116.
from t = 2 to t = 2.5
117.
from t = 2.5 to t = 2.8
118.
from t = 2.8 to t = 3
(4.2, 2.8) (5.3, 0) t 119.
from t = 3 to t = 3.9
120.
from t = 3.9 to t = 4.2
121.
from t = 4.2 to t = 5.8
122.
What is the farthest distance that Kevin is from home?
The following is a graph that represents the speed v (in miles per hour) of Michael as a function
of time t (in minutes).
(7, 50) (2, 30) (8, 38) (4, 30) (4.2, 0) (7.4, 50) (7.6, 38) (6, 0) (9.1, 0) 123.
Over what interval of time is Michael traveling fastest?
124.
Over what interval(s) of time is Michael’s speed zero?
125.
What is happening to Michael’s speed between 0 and 2 minutes?
126.
Describe Michael’s speed between 7 and 7.4 minutes.
127.
How does the answer in #126 compare to that of #124?
128.
During what interval(s) is Michael’s speed constant?
t