Summer Work
Pre-Calculus
Instructions:
1. Do all work on a separate piece of paper.
2. Make sure you clearly write the question number and the title of the
section.
3. Show all work necessary. Just answers are not enough.
4. All work should be finished before school starts. You will submit this work
as soon as you are back in fall.
5. If you work with others (highly encouraged), do not copy their work.
6. The ultimate goal is to help you review, if you cheat, you might get the
points, but you will not be ready for next year.
By signing below, I confirm that I have completed all this work without copying or cheating from
anyone else.
Name:________________________
Signature:_____________________________
Quadratic Functions
1. Sketch the graph of the given functions and write the Domain, Range, Vertex, X-intercept, Yintercept, and the Line of Symmetry
a. 𝑓 𝑥 = 2𝑥 ! − 12𝑥 + 14
b. 𝑓 𝑥 = 2 𝑥 − 4 ! + 2
c. 𝑓 𝑥 = 3(2𝑥 − 1)(𝑥 + 4)
2. Solve each equation by either using factoring, completing the square or the quadratic formula.
a. 𝑥 ! − 14𝑥 − 32 = 0
d. 5𝑥 ! − 10𝑥 − 15 = 0
b. 𝑥 ! = −2𝑥
e. 3𝑧 ! + 4𝑧 + 5 = 0
!
c. 3𝑎 − 18 = 3𝑎
f. 𝑥 ! + 2𝑥 − 13 = 0
3. What is the discriminant of a quadratic equation and how do we use it find the types of solutions?
4. You throw a baseball up in the air with an upward initial velocity of 48ft/s. The height, h (in feet),
above the ground of the ball is modeled by the function ℎ 𝑡 = −16𝑡 ! + 48𝑡 + 3. Answer the
following questions for this situation:
a. What is the initial height of the ball?
b. If no one catches the ball, when does it land on the ground?
c. What is the maximum height of the ball?
d. At what time does the ball reach its maximum height?
e. If a catcher catches the ball 3 ft above the ground, how long was the ball in the air? Why do
you get two answers? Which one is more reasonable?
Polynomial Functions
5. Use long division to simplify: (𝑥 ! + 𝑥 ! − 5𝑥 ! + 𝑥 − 6) ÷ (𝑥 + 3). Clearly write the quotient and
the remainder.
6. Use synthetic division to simplify: (𝑥 ! + 𝑥 − 5) ÷ (𝑥 − 3)
7. Find all solutions (rational, irrational, or complex):
a. 0 = 𝑥 ! + 𝑥 ! − 2
b. 36 = 4𝑥 ! + 16𝑥 ! − 9𝑥
8. Write a polynomial function with the given characteristics. Multiply out the polynomials in parts a
and b, but leave part c in factored form.
a. Third Degree (cubic), has x-intercepts at (3,0) and (−2,0) with a multiplicity of 2.
b. 4th degree polynomial with zeros at 2, −4 and 3.
c. An 8th degree polynomial with zeros at 1(multiplicity of 3), -1 (multiplicity 2), 4(multiplicity
1), and – 𝑖(multiplicity of 1).
9. Find the following information for the function: 𝑓 𝑥 = 𝑥 ! − 11𝑥 ! + 16𝑥 ! − 12𝑥 + 16.
a. What is the maximum number of zeros this function can have?
b. How many times can the function turn?
c. What is the end behavior of this function?
d. What is the y-intercept?
e. What is 𝑓 2 =?
f. Find all x-intercepts and the multiplicity of each.
g. How does the multiplicity help?
h. Find all non-real zeros.
i. Sketch the graph. Use the graphing calculator to check your work.
Exponential and Logarithmic Functions
10. For each of the following functions, sketch the graph and write the Domain, the Range, the
Asymptote, the x-intercept, the y-intercept and whether graph represents exponential or logistic
growth or decay.
a. 𝑓 𝑥 = 2! − 4
d. 𝑓 𝑥 = log ! 𝑥 + 1
! !
e. 𝑓 𝑥 = 1 + log ! 𝑥
b. 𝑓 𝑥 =
+2
!
c. 𝑓 𝑥 = 0.5
!
f. 𝑓 𝑥 = − log ! 𝑥
!!!
11. Solve each of the given equations and check for extraneous solutions. No calculators allowed.
a. 9! = 27
b.
c.
d.
e.
f.
!
!
!"
!
=4
!
3 =!
log ! 𝑥 = 2
log(𝑥 − 1) = 1
log ! 𝑥 + log ! 𝑥 + 1 = 1
12. You buy a car for $30,000. A car depreciates in value at 17% per year. How much will this car be
work in 5 years?
13. You want $200,000 in 10 years. You find an account that pays 2.5% compounded interest
compounded monthly. How much should you invest in this account?
14. You want to double your money in an account that pays 4.1% compound interest compounded
continuously. How long will this take?
Radical Functions
15. Write each of the given in radical form.
a.
7𝑎
!
!
!
b. 𝑝!!
16. Write each of the given in radical form.
a.
!
4𝑥
17. Simplify.
!
!
!
a.
64𝑏 !
b.
125𝑥 !
!
!
!
!
!
c.
49𝑝!
d.
𝑐 ! 𝑑! 𝑓
!
! !
!
b.
e.
f.
g.
!
!
!
!"!
!
! !
𝑥 !" 𝑦 ! 𝑧 !
! !
!"
! !!
𝑏
𝑎
! ! !
𝑎𝑏 𝑐
𝑥 ! 𝑦 ! 𝑧!"
!
!
c.
10𝑥
c.
𝑝 8
!!
h.
!
!
!
!
!
!
!!!
! !! ! !!
i.
40 + 4 5 − 2 3
j. 4 6 ∗ −2 3
18. Solve each of the given equations. Check for extraneous solutions.
b. 2𝑥 + 18 = 𝑥 + 3
a. 5𝑟 = 4𝑟 + 2
19. Graph the given functions and write the Domain, the Range, the x-intercept and the y-intercept:
!
b. 𝑓 𝑥 = 𝑥 + 1
a. 𝑓 𝑥 = −2 𝑥 − 5 + 3
Rational Functions
20. Simplify and write all necessary restrictions:
a.
b.
! ! !!!!"
÷
! ! !!"!!!"
!
!
!
!!
!
!!
! ! !!!!!"
c.
! ! !!!!!
!
!!!
!
− !!!
!
!
d. − !!! + !! !!
21. Solve the given equations and check for extraneous solutions:
!
!
!!
e. !! ! !!!!!" + !! ! !!!!!" = !! ! !!"!!!"
f.
!! !
! ! !!
!!
!
− !!! = !!!
22. Sketch the graph of the given function and write the domain, range, x and y intercepts, horizontal
and vertical asymptotes.
𝑥 ! − 2𝑥 − 8
𝑓 𝑥 =
9 − 𝑥!
Trigonometric Functions
23. For each of the given angle, convert to radian or degrees, sketch the angle and write two co-terminal
angles:
!!
c. 𝜃 = 4.93 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 a. 𝜃 = 560!
b. 𝜃 = −
!
24. Find the value of the following without a calculator:
!
!
a. cos − !
d. sec − !
b. csc
!!
!
c. sec 405!
e. sin
!!!
!
f. sin −135!
g. tan 225!
25. Identify the quadrant with the given characteristics:
a. sin 𝜃 < 0, cos 𝜃 > 0
b. tan 𝜃 > 0, csc 𝜃 < 0
h. cot(−300! )
!!
i. tan − !
j. csc
!!
!
c. sec 𝜃 < 0, cot 𝜃 < 0
d. cos 𝜃 > 0, tan 𝜃 > 0
26. Given the point −3,4 is on the terminal side of some angle 𝜃, find all six trigonometric ratios for 𝜃.
27. Sketch the graph of 𝑓 𝑥 = sin(𝑥), 𝑓 𝑥 = cos(𝑥) and 𝑓 𝑥 = tan(𝑥) and write the domain, range,
amplitude and period of all.
28. Find all the missing parts of the given triangles. You may have to use a calculator.