Study Guide for College Algebra Final
Simplify radicals and exponential expressions.
3
1. √120𝑎6 𝑏13
3.
2 x
2

3
y 4 5 x3 y

3
2. √384𝑎6 𝑏13

4. 5a 2 b 7
 7ab 
3
4 2
Simplify a rational expression.
5.
𝑥 2 −16
6𝑥+12
×
3𝑥+6
𝑥 2 +3𝑥−4
6.
2𝑥 2 +16𝑥+5
𝑥 2 −9
÷
4𝑥 2 +16𝑥+16
𝑥 2 −𝑥−6
Model a linear equation
7. John bought a car at a cost of $2,000 in 2005. Assume the car loses its value at a constant rate over
time. If it was worth $1,200 in 2008, how much was it worth in 2012?
Find the equation of a circle and locate the radius and center.
8. Find the radius and center of the circle 𝑥 2 + 𝑦 2 − 6𝑥 + 10𝑦 + 18 = 0.
Given a radius and center, find the equation of the circle.
9. Find the equation of the circle with center at (-1, -4) and radius 12.
Simplify complex numbers and negative square roots.
10. (4 − √−3)(2 + √−27)
11. Multiply (2 − 3𝑖) by its conjugate.
Solve equations that include linear, quadratic, rational, radical, factoring, and higher degrees.
12. √2𝑥 + 12 + 2 = 𝑥
13. 𝑥 2 − 3𝑥 = −10
14. 3𝑥 4 − 2𝑥 3 + 𝑥 2 = 0
15.
16. 36𝑥 4 − 49 = 0
17. −3(4𝑥 − 2) + 3𝑥 = 2 − (5𝑥 + 3)
𝑧+1
𝑧+3
=
4
𝑧−3
+
16
𝑧 2 −9
Given a graph, find intervals of increase and decrease, <, > 0, find values given information, minima and
maxima, degree, positive or negative functions, x-intercepts with multiplicity, y-intercepts, power functions,
factored form, and asymptotes.
18. Given the graph of f(x) to the right, solve the following.
a. f(-5) = ____
b. f(2) = ____
c. For what x is f(x) = -2? ________________
d. On what intervals is f(x) increasing? Decreasing?
e. What is the domain of f(g)? Range?
f. Is f(x) a function? Why or why not? Is it one-to-one? Why or why not?
g. On what intervals is f(x) > 0? On what intervals is f(x) < 0?
h. Local maxima? Local minima?
19. Given the graph of g(x) to the right, solve the following.
a. g(-1) = ____
b. g(0) = ____
c. For what x is g(x) = 0? ________________
d. On what intervals is g(x) increasing? Decreasing?
e. Is g(x) positive or negative? Is g(x) of odd or even degree?
f. On what intervals is g(x) > 0? On what intervals is f(x) < 0?
g. What are the factors of g(x)? Include multiplicity.
h. What is the end behavior of g(x)? As 𝑥 → −∞, 𝑦 →_____. As 𝑥 → ∞, 𝑦 →____.
Find specific piecewise functions. Graph a piecewise and figure out the piecewise given the graph including
the domain and the functions.
20. Write the piecewise function f(x) described by the graph.
Evaluate the following:
f(-2) = ____
f(0) = ____
f(1) = ____
f(3) = ____
21. Graph the following piecewise function:
𝑔(𝑥) = {
−3
|𝑥 − 1|
𝑥 ≤ −2
𝑥 > −2
Evaluate the following:
g(-4) = ____
g(-2) = ____
g(0) = ____
Given a function, find the degree, the end behavior, the x-intercepts and their multiplicity and behavior, the
y-intercept, and the power function.
22. Given the function 𝑔(𝑥) = −2𝑥(𝑥 − 3)2 (𝑥 + 2)3 , answer
the following questions and create a graph.
a. What are the x-intercepts? What is their multiplicity
and their behavior?
b. What is the y-intercept?
c. What is the power function?
d. As 𝑥 → −∞, 𝑦 →_____. As 𝑥 → ∞, 𝑦 →____.
e. Is g(x) of even or odd degree?
Given a quadratic function, find the vertex, minimum or maximum, intercepts, concavity, and create a graph
using the found information. An additional revenue problem will be added.
23. An arrow is shot from a height of 5 feet. The height of the arrow (in feet) above the ground at time t
seconds, after it is shot is given by the equation 𝑦 = −16𝑡 2 + 96𝑡 + 5.
a. At what time will the arrow reach its maximum height?
b. What is the maximum height?
c. When will the arrow hit the ground?
d. Can the arrow reach a height of 200 feet? Will it reach 500 feet? Explain.
24. Given the quadratic 𝑔(𝑥) = −3𝑥 2 + 3𝑥 + 18, answer the following.
a.
b.
c.
d.
e.
What is the concavity of the parabola? Why?
Find the vertex. Is it a maximum or a minimum? Why?
Find the y-intercept
Find the x-intercepts.
Graph the parabola on the right.
Graph rational functions including asymptotes and intercepts. Solve inequalities using test points.
25. Given the function ℎ(𝑥 )
a.
b.
c.
d.
e.
=
𝑥 2 −𝑥−20
𝑥+7
, answer the following:
Vertical asymptotes:
Horizontal asymptotes or slant asymptotes:
y-intercepts
x-intercepts
Domain in set notation
26. Given the function 𝑔(𝑥 )
a.
b.
c.
d.
e.
=
𝑥 2 −9
, answer the following:
2𝑥 2 −8
Vertical asymptotes:
Horizontal asymptotes:
y-intercepts
x-intercepts
Domain in set notation
27. Solve the inequality
3
𝑥+1
≤ 2 using test points.
List all transformations and be able to graph a given function.
28. Write an equation for a function h(x) that has the shape of
𝑗(𝑥) = 𝑥 5 but is shifted right 6 units and down 4 units.
29. Sketch a graph of the function f(x) to the right which has been
up 5, shifted left 4, stretched by a factor of 2 units, reflected
across the x-axis, reflected across the y-axis.
30. List the transformations used to graph the
function
𝑓(𝑥) = −2√𝑥 + 1 + 3. Use those
transformations to
create the graph on the right.
31. List the transformations used to graph the
function
𝑓(𝑥) = 3 + 𝑒 −𝑥+1 . Use those transformations
to
create the graph on the right. Include any
asymptotes.
List the transformations used to graph the function
𝑓(𝑥) = 2 − log(𝑥 − 1). Use those transformations to create the graph on the
right. Include any asymptotes.
Find the domain and range of a function, decide if something is a function and if it is one-to-one, combine
functions, use composition with functions, find the inverse of a function, and simplify the difference
quotient.
32. Given 𝑓(𝑥) = 3𝑥 − 5, 𝑔(𝑥) = √2𝑥 + 4, 𝑘(𝑥) = 𝑥 2 + 6, and 𝑗(𝑥) =
a.
b.
c.
d.
2𝑥+1
, answer the following.
√𝑥−3
Find the domain of each function.
Find 𝑓 −1 (𝑥) and 𝑔−1 (𝑥).
Find 𝑔(𝑓(𝑥)) and 𝑘𝑔(𝑥)). Find 𝑔(𝑗(4)) and 𝑘(𝑓(2)).
Which functions are one-to-one? How do you know?
𝑓
𝑘
𝑘
𝑔
e. Find ( ) (𝑥) and ( ) (𝑥). Write their domains in set notation.
f. Find
𝑘(𝑥+ℎ)−𝑘(𝑥)
ℎ
.
Find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
.
Go back and forth between exponentials and logarithms, solve both types of equations including word
problems, simplify logs with and without a calculator, combine and separate logs, use change of base, and
solve log and exponential equations.
33. Rewrite the exponentials as logarithmic equations and solve for x:
a. 72𝑥+3 = 12
b. 2𝑥+1 = 16
c. 25𝑥+7 = 125
34. Rewrite the logarithmic equations as exponentials and solve for x:
a. 𝑙𝑜𝑔𝑥 10 = 4
b. 𝑙𝑜𝑔6 36 = 𝑥
c. 𝑙𝑜𝑔5 7 = 𝑥
35. Rewrite the expression 𝑙𝑜𝑔5 𝑥 − 4𝑙𝑜𝑔5 𝑧 as a single logarithm.
𝑥𝑦 3
36. Rewrite 𝑙𝑜𝑔2 (
) as separate logarithms.
𝑧
37. Use the properties of logarithms to solve for x:
a. 𝑙𝑛(𝑥 − 1) + 𝑙𝑛(𝑥 + 3) = 𝑙𝑛(2)
b. 𝑙𝑜𝑔4 (𝑥 2 − 1) − 𝑙𝑜𝑔4 (𝑥 + 1) = 0
38. Solve the exponential equations for x:
200
a.
= 50
b. 8 = 11 − 2𝑒 −.3𝑥
1+5𝑒 2𝑥
39. The value of a stock is decreasing exponentially at a rate of 14% per hour.
a. If the value of the stock started at $2,000, model the information as an exponential.
b. Using your equation in part a, how much is the stock worth after 4 hours?
c. How many hours will it take for the value of the stock to cut in half?
40. The value of a new stock is increasing exponentially at a rate of 10% per hour.
a. If the value of the stock started at $2,000, model the information as an exponential.
b. Using your equation in part a, how much is the stock worth after 4 hours?
c. How many hours will it take for the value of the stock to triple?
41. How long will it take for the population of a certain country to triple if its annual growth rate is 5.4%
and the population is currently 3 million? Round to the nearest year.
42. The number of acres in a landfill is given by the function B  8100e 0.04t where t is measured in years.
How many acres will the landfill have after 2 years? (Round to the nearest acre.)
43. The sales of a mature product begin to decline at an exponential rate given by the function
S (t )  S o e  kt where t is time in years.
a. If sales have declined from 69,400 to 1553 units after 20 years model the decline in sales in
terms of S (t )  S o e  kt .
b. What are the sales after 12 years.