College Prep
Chapter 3 Review
Name_______________________
1. Complete the square to write the function in shifted form.
a. 𝑓(𝑥) = 𝑥 2 + 7𝑥 + 4
b. 𝑓(𝑥) = 4𝑥 2 − 24𝑥 + 15
2. Carefully graph the function, stating end behavior, vertex, and intercepts (as points). (Use the
quadratic formula to find the x-intercepts rounded to one decimal place if needed.)
𝑓(𝑥) = 4𝑥 2 − 8𝑥 + 2
end behavior ___________________
vertex: _________________
x-intercepts: ___________
___________
y-intercept: ____________
3. Carefully graph the function, stating end behavior, vertex, and intercepts (as points). (Use the
quadratic formula to find the x-intercepts rounded to one decimal place if needed.)
𝑓(𝑥) = −3𝑥 2 − 6𝑥 + 6
end behavior ___________________
vertex: _________________
x-intercepts: ___________
___________
y-intercept: ____________
4. Use the remainder theorem to determine if each is a factor of 𝑝(𝑥) = 𝑥 4 + 𝑥 3 − 10𝑥 2 − 4𝑥 + 24
(show appropriate work)
a. (𝑥 − 2)
b. (𝑥 + 1)
5. Find a cubic polynomial, 𝑝(𝑥), with real coefficients having roots 𝑥 = 3 and 𝑥 = −2𝑖. Assume a lead
coefficient of 1.
6. Find a fourth degree polynomial, 𝑝(𝑥), with real coefficients having roots 𝑥 = −3, 𝑥 = 1 and 𝑥 = 4𝑖.
Assume a lead coefficient of 1.
7. Use the rational roots theorem to list all possible rational roots of 3𝑥 4 + 14𝑥 3 − 𝑥 2 − 42𝑥 − 24 = 0
8. Use the rational roots theorem to write the function in factored form and find all zeroes.
a. 𝑝(𝑥) = 𝑥 4 − 10𝑥 3 + 90𝑥 − 81
b. 4𝑥 4 − 15𝑥 3 + 9𝑥 2 + 16𝑥 − 12
9. State the degree, end behavior, and intercepts for the function. Indicate the graph’s behavior at each xintercept by circling b (bounce) or c (cross).
a. 𝑓(𝑥) = (𝑥 − 1)5 (𝑥 + 7)(𝑥 + 2)2
degree ____________ end behavior ______ / ______
x-intercepts: ___________ ___________ ___________
b or c
b or c
b or c
y-intercept: ____________
b. 𝑓(𝑥) = (𝑥 + 4)2 (𝑥 − 2)2 (𝑥 − 1)
degree ____________ end behavior ______ / ______
x-intercepts: ___________ ___________ ___________
b or c
b or c
b or c
y-intercept: ____________
10. Carefully graph the polynomial. State end behavior and find all intercepts.
a. 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 11𝑥 + 12
end behavior: ________ / ________
x-intercepts: ____________________
___________________
___________________
y-intercept: _____________________
b. 𝑓(𝑥) = 𝑥 3 − 21𝑥 + 20
end behavior: ________ / ________
x-intercepts: ____________________
___________________
___________________
y-intercept: _____________________
11. State the domain and range of the functions.
12. Determine the intercepts and asymptotes.
𝑓(𝑥) =
4𝑥 2 −16
𝑥 2 −36
y-intercept: ____________
x-intercepts: ___________
___________
vertical asymptotes: ___________
___________
horizontal asymptote: __________________
13. Carefully graph the rational function. State the intercepts and asymptotes.
𝑥−4
𝑓(𝑥) = 𝑥+3
y-intercept: ____________
x-intercept: ___________
vertical asymptote: ___________
horizontal asymptote: __________________
14. Determine the oblique asymptote.
a. 𝑓(𝑥) =
𝑥 3 −𝑥 2 −9𝑥+9
𝑥2
b. 𝑓(𝑥) =
𝑥 2 +3𝑥+4
𝑥
15. If there is a removable discontinuity, repair the break by redefining the function using an appropriate
piecewise-defined function.
a. 𝑓(𝑥) =
𝑥 2 −2𝑥−3
𝑥+1
16. Solve the inequality using the graph. 𝑓(𝑥) ≤ 0
b. 𝑓(𝑥) =
𝑥 2 −9
𝑥+3
17. Solve the inequality by finding zeroes and
testing intervals. 𝑥 2 + 4𝑥 − 5 < 0
18. Solve the inequality by finding zeroes and excluded values and testing intervals. Answer in interval
notation.
𝑥2 − 9
≥0
𝑥 2 − 12𝑥 + 11