College Algebra
Name
Final Exam Study Guide
Directions: Show all work and reasoning to receive full credit.
UNIT 1 (A.8, A.9, 1.1, 1.4, 1.5)
For #1 – 6, simplify the expressions.
1)
4√24+3√32−2√50
2
2)
−2
3
√9
3)
2−√2
3
3
4) √15𝑥 5 ∙ √9𝑥 4 𝑦 7
√2+3√3
For #7 – 10, solve the inequalities. Write answers in interval notation and graph on a number line.
−4𝑥−3
7) 𝑥(4𝑥 + 3) ≤ (2𝑥 + 1)2
8) −9 <
9) 6 − 3|1 − 2𝑥| − 4 ≤ 11
10) |2 − 3𝑥| < |−7| − 6
3
≤3
11) Plot the points A(-2, 5), B(12, 3), and C(10, -11). Sketch triangle ABC and verify that the triangle is a right
triangle. Determine the area of triangle ABC.
12) Find all the points on the x-axis that are 5 units from the point (4, -3)
1
13) If a line segment has one endpoint at (-4, 3) and a midpoint at (−1, − 2), find the other endpoint.
14) Determine the standard form and the general form of the equation of a circle with radius r = 4 and
center (h, k) = (-2, 1). Sketch the circle on the coordinate plane.
15) Determine the standard form and the general form of the equation of a circle with center (1, -3) that includes
the point (5, -5). Then find the intercepts.
16) Determine the center (h, k) and radius r of the circle 4 x2  4 y 2 12 x  16 y  5  0 . Graph the function.
1
1
For Questions 17 & 18, determine an equation for the line with the given properties.
17) Parallel to the line 4 x  3 y  3 ;
18) Perpendicular to the line 2 x  9 y  66 ;
containing the point (6, 0).
containing the point (6, -13).
19) Find the equation of the line in slope-intercept form that connects the centers of the two given circles.
x 2  y 2  4 x  8 y  10  0
and
x 2  y 2  18 x  18 y  59  0
20) Determine the general equation of the circle shown in the figure below if the midpoint of line segment L is
the center of the circle.
UNIT 2 (2.1-2.7)
For Questions 1 & 2, determine the domain of the functions. Answers should be in interval notation.
x4
1) h ( x )  3
2) g ( x )  8  22 x
2 x  50 x
3) For the given functions f and g, find the following functions and the domain of each:
𝑓(𝑥) =
a) f – g
2𝑥 + 3
;
3𝑥 − 2
𝑔(𝑥) =
4𝑥
3𝑥 − 2
b) f  g
4) Determine and simplify the difference quotient f,
c) f g
f ( x  h)  f ( x )
, h  0 , given f ( x)  2 x2  9 x  5 .
h
5) Answer the following questions about the function f (x) 
3x  6
x2  4
5
a) If f (x)  , what is x? What are the point(s) on the graph?
3


b) List the x-intercepts and y-intercept, if any, of the graph of f. Check your answers.
For Questions 6 & 7, determine algebraically whether the function is even, odd, or neither. Then state any
symmetry.
 x3
3
6) f ( x)  2
7) g ( x)  3 x 2  8
4x  9
8) Use a graphing utility to graph the function f ( x)  2 x 3  7 x 2  3x .
a) Determine to the nearest thousandths any relative minimum and maximum values.
b) Determine the open interval(s) on which the function is increasing, decreasing, or constant.
9) Use the graph to the right to answer the following questions a - f.
a) Determine the interval when the function is increasing.
b) Determine the interval where the function is constant.
c) Determine the interval where the function is decreasing.
d) List the intercepts (x, y) for the function.
e) Find f (3).
f) For what value of x does f (x) = -2?
10) Find the equation of the secant line in slope-intercept form for the function f (x) 
points (-1, f (-1)) and (2, f (2)). No decimal approximations.

x3
containing the
x2  3
11) Use the function f ( x )  2 x 2  x to answers questions a and b.
a) Find and simplify the average rate of change from -1 to x.
b) Find the equation of the secant line in point-slope form containing (-1, f (-1)) and (4, f (4)).
12) At 2:00 p.m. there was 9 feet 8 inches of snow at Breckenridge Ski Resort. By 10:00 p.m. the same night
there was 10 feet 3 inches of snow. What is the average rate of change (in inches) for the snow level?
13) Write a rule for the piecewise function graphed.



f ( x)  



1
1
15) Graph by hand the following function.
List the sequence of transformations.
1
f ( x)  4 x  2  5
2
1
14) Graph the given piecewise function.
 1
 2 x  2, (, 2]

 1
f ( x)   x 2  3, (2, 4]
 2
 3x  13, (4, )


1
16) Graph by hand the following function.
List the sequence of transformations.
g ( x )  3 2 x  8  7
1
1
1
1
17) Determine the equation of the following graph.
1
18) Determine the equation of the following graph.
1
1
1
19) A wire of length 5x is bent into the shape of a square. Express the Area as a function of x.
20) A marina owner wishes to estimate a linear function that relates boat length in feet and its draft (depth of
boat below water line) in feet. He collects the following data. Let boat length represent the independent
variable and draft represent the dependent variable. Use a graphing utility to draw a scatter plot and to find the
line of best fit. Use the equation to determine the draft for a boat 60 ft in length (to the nearest thousandth).
Boat Length (ft)
25
25
30
30
45
45
50
50
Draft (ft)
2.5
2
3
3.5
6
7
7
8
21) An open box is made from a rectangular piece of material by cutting equal squares from each corner and
turning up the sides.
a) Write the volume of the box as a function of x if the
material is 24 inches by 16 inches (use function notation).
Simplify and write your equation in standard form.
b) Graph the function on your calculator. Determine the
value of x that will maximize the volume of the box.
22) Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The
company chooses to depreciate the machine using the straight-line method over 12 years.
a) Write a linear function that expresses the book value of the machine as a function of its age.
b) What is the book value of the machine after 30 months?
c) When will the machine be worth $45,000? (show work)
UNIT 3 (3.1-3.7)
1) Use the quadratic function g ( x)  16 x2  32 x  15 to (no need to complete the square):
a) Find the axis of symmetry.
b) Find the vertex.
c) Find the x- and y-intercepts.
2) Algebraically determine the axis of symmetry, vertex, x-intercepts (in radical form), y-intercept, domain,
range, and intervals of increasing/decreasing of the following quadratic function by completing the square and
writing it in transformation form.
f ( x)  2 x 2  10 x  3
3) Determine an equation in transformation form for the graph of the parabola.
Write the equation in standard form.
20
18
16
14
12
10
8
6
4
2
-12 -10 -8 -6 -4 -2
-2
2 4 6 8 10 12
-4
4) The following table shows the median number of hours of leisure time that Americans had each week in
various years.
Year
Median # of Leisure hrs per Week
1973
26.2
1980
19.2
1987
16.6
1993
18.8
1997
19.5
Use x = 0 to represent the year 1973. Using a graphing utility, determine the quadratic regression equation for
the data given (round to the nearest thousandths). Using the quadratic equation,
Equation:
a) determine the year that corresponds to the time when Americans had the least time to spend on leisure.
b) determine the minimum median number of hours spent on leisure.
5) A suspension bridge has twin towers that are 1300 feet apart. Each tower extends 180 feet above the road
surface. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the
road surface at the center of the bridge. Illustrate this scenario on a coordinate plane and find the height of the
cable at a point 200 feet from the center of the bridge.
6) A projectile is fired from a cliff 300 feet above the water and an inclination of 45o to the horizontal, with a
muzzle velocity of 250 feet per second. The height h of the projectile above the water is given by
32 x 2
h( x ) 
 x  300 , where x is the horizontal distance of the projectile from the base of the cliff. How far
(250)2
from the base of the cliff is the height of the projectile a maximum and what is this maximum height?
7) For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches
the x-axis at each x-intercept and state the end behavior.
f ( x )   x 2 ( x  3)3 ( x  7) 4
8) Form a polynomial whose zeros and degrees are given.
Zeros: 3, multiplicity 2; 0, multiplicity 3; -2, multiplicity 2
9) Analyze the graph of the given function f as follows: f ( x)  ( x  3)( x2  4)( x  6)3
a) Determine the degree of the function.
b) Determine the end behavior.
c) Find the x- and y- intercepts of the graph.
d) Identify the multiplicity of each zero and state whether it crosses or touches the x-axis.
e) Graph f using a graphing utility and determine the local maxima and local minima, if any exist (round
to nearest thousandths).
f) Determine the domain and range.
g) Determine the intervals on which the graph of f is increasing and decreasing.
1) Accurately graph the following function by determining the following: f ( x) 
a) horizontal or oblique asymptote
3x 2  11x  4
2 x 2  5 x  12
b) write f in lowest terms
1
1
c) vertical asymptotes and holes
e) x- and y-intercepts
d) domain and range
f) additional points
2) Accurately graph the following function by determining the following: f ( x) 
a) horizontal or oblique asymptote
x3  x 2  12 x
x2  4
b) write f in lowest terms
1
1
c) vertical asymptotes and holes
e) x- and y-intercepts
d) domain and range
f) additional points
3) While traveling in a car, the centrifugal force F a passenger experiences as the car drives in a circle varies
jointly as the mass m of the passenger and the square of the speed v of the car. If a passenger experiences a
force of 144 newtons when the car is moving at a speed of 40 kilometers per hour and the passenger has a mass
of 100 kilograms, find the force a passenger experiences when the car is moving at 70 kilometers per hour and
the passenger has a mass of 80 kilograms.
4) The time t in hours it takes a satellite to complete an orbit around the earth varies directly as the radius r of
the orbit (from the center of the earth) and inversely as the orbital velocity v. If a satellite completes an orbit
730 miles above the earth in 10 hours at a velocity of 29,000 mph, how long would it take a satellite to complete
an orbit if it is at 1200 miles above the earth at a velocity of 30,000 mph? (Use 3960 miles as the radius of the
earth and round final answer to the nearest thousandths.)
5) Solve the following inequalities.
a) 4 x3  4 x  6 x 2
b) 2 x5  5 x 4  18 x3  45 x 2
c)
12 x
 6x
4 x
6) Determine all of the complex zeros of the given polynomial functions (no decimal answers).
a) g ( x)  2 x2  x  5
b) h( x)  4 x4  22 x2  42
c) f ( x)  3x4  81x
7) Determine all of the real zeros of the polynomial function, and then use the real zeros to factor f over the real
numbers.
f ( x)  3x4  x3  15x 2  x  2
8) Find all of the complex zeros of the polynomial function and write the polynomial as a product of linear
factors.
f ( x)  4 x4  9 x3  21x2  36 x  20
9) Use the given zeros to find the remaining zeros of the function. No decimal values.
f ( x)  3x4 19 x3  69 x2  99 x  26; zero: 2  3i
10) Find all zeros of the function and write the polynomial as a product of linear factors (no decimal values).
g ( x)  x4  2 x3  5x2 18x  36
11) Find a fourth degree polynomial in standard form that has the given zeros: 1, 3, 2  3i