MATH 1113 College Algebra
Final Exam Review
I would recommend that you also study all of your old exams, worksheets, and notes in addition
to completing this review.
Section 1.1
1. Determine whether or not each of the given tables represents y as a function of x.
a)
b)
x -1 0 1 2 3
x 5 7 -1 5 -8
y 5 7 -1 5 -8
y -1 0 1 2 3
2) Tell whether or not each set of ordered pairs defines y as a function of x. Then state the
domain and range of each.
a) {( 5,1), (3,2), (4,9), (7,6)}
b) {( 2,4), (0,2), (2,5)}
3) Determine whether or not each relation defines y as a function of x.
a)
b)
c)
d) y  2 x  3
e)
y  3x 2  5
f) x  3 y 2  4
g) x 2  y 2  25
Section 1.2
4) An electric utility company determines the monthly bill by charging 10.5 cents per kilowatt-hour
(KWH) used plus a base charge of $5.80 per month.
a) Write a linear function that models this data, where W is the number of KWH and cost is f (W ) .
b) Find f (1000) and explain what it means in the context of this situation.
c) What is the monthly charge if 1500 KWH are used?
5) The cost of prizes & expenses of state lotteries is given by L  35t 2  740t  1207 million dollars,
with t equal to the number of years after 1980.
a) What are the values of t that correspond to the years 1982, 1988, and 2000?
b) L  f ( 4) gives the value of L for what year? What is f (4) ?
c) What xmin and x max should be used to set a viewing window so that t represents 1980 – 1997?
6) The rate (number per 100,000 people) of juvenile arrests is given by
f ( x)  0.027 x 2  5.69 x  51.15 , where x is the number of years after 1950.
a) Use technology to graph this model on the viewing window [0, 50] by [0, 300].
b) This viewing window shows the graph for what time period?
c) Did the rate of juvenile arrests increase or decrease over this period?
d) Use the model to find the rate of juvenile arrests in 1960 & in 1998.
Sections 1.3 & 1.4 (and more)
7) How can you identify a) a linear equation? b) a quadratic equation? c) a cubic equation?
d) an exponential equation?
8) State the slope of each line whose equation is given, and graph the line.
1
y3
2
9) Write an equation in slope-intercept form for each line described. Then give the slope and
y-intercept and graph.
a) y  54 x
b) 3x  2 y  8
a) through (2, 4), m = –1
b) through (8, –1) and (4, 3)
c) vertical, through (–6, 4)
e) through (1, 2); parallel to y  2 x  3
d) x intercept –2, y intercept 4
5
f) through (2, –1); perpendicular to y   x  2
3
h) through (–1, 3 ) and ( 3, 4)
d) 2 x 
c) y  4
g) through ( 1,3 ) with slope – 2
i) horizontal through (–2, 7)
10) Write a linear function that computes the long distance phone bill in dollars for calling t minutes
at 20 cents per minute plus a fixed fee of $3.95 each month.
11) Assume that the growth of the membership of a country club was linear from 1996 to 2000 with a
membership of 181 in 1996 and a rate of growth of 699 members per year. Write an equation for
the membership M of the country club as a function of the number of years t after 1996. Then use
the function to approximate the membership in the year 2000.
12) A business uses straight-line depreciation to determine the value V of an automobile over an 8
year period. Suppose the original value (when t = 0) is $23,500 and the salvage value (when t = 8)
is $5000.
a) By how much has the automobile depreciated over the 8 years?
b) How much is the automobile depreciating each year? Give units!
c) Write the linear equation that models the value V of this automobile at the end of year t.
13) A company finds that it can produce 25 heaters for $6500, while producing 45 heaters for $9000.
Express the cost y as a linear function of the number of heaters x.
Section 2.1
14. Solve each of the following linear equations.
5x  2 1
1
3 1
1 4
 x  x
  x
a)
b)
4 5
2 5
5
2
10
c)
5
1 2
x  2x  
6
3 3
Section 2.3
15) Solve each of the following linear systems algebraically. If there are many solutions, say so.
If there is no solution, say so. Otherwise, state both coordinates of the solution.
a)
3x  4 y  4
x  y  13
b)
x  5y  8
x  6y
c)
4 x  y  23
x  2 y  17
d)
x y 7
2 x  2 y  14
Section 2.4
16) A company is producing a new logic board for computers. The annual fixed cost for the board is
$435,000 and the variable cost is $150 per board. If the logic board sells for $500, write and solve
an inequality that gives the number of logic boards that will give a profit for the product.
17) It is estimated that the annual cost of driving a certain new car is given by the formula
C  0.35m  2200 , where m represents the number of miles driven per year and C is the cost in
dollars. Jan has purchased such a car, and decides to budget between $6400 and $7100 for next
year’s driving costs. What is the corresponding range of miles that she can drive and still stay
within her budget?
Section 3.1
18) The graph of each equation below is a parabola. State the vertex of each parabola.
1
a) f ( x)  ( x  5) 2  2
b) f ( x)  2( x  8) 2  7
3
c) y  2 x 2  12 x  7
d) y  x 2  8 x  1
19) The profit for a product can be described by the function P( x)  40 x  3000  0.01x 2 dollars,
where x is the number of units produced and sold. To maximize profit, how many units must be
produced and sold? What is the maximum possible profit?
20) The annual total revenue for a product is given by R( x)  1500 x  0.02 x 2 dollars, where x is the
number of units sold. To maximize the annual revenue, how many units must be sold? What is
the maximum possible annual revenue?
Section 3.2
21) Solve each of the following quadratic equations. You may use the zero-factor property, square
root property, completing the square, and/or the quadratic formula.
a) x 2  2 x  8
b)  4 x 2  x  3  0
c) ( x  4) 2  20
d) x 2  7 x  12
e) 3 x 2  2 x  5  0
f) 4 x 2  10  3 x
g) x 2  2 x  10
h) 3 x 2  x  10
Section 3.3
22) Given
for
x  3
1  x

f ( x)  3
for  3  x  2
 2 x  2 for
x2

(a) f (4)
(b) f (2)
find each of the following:
(c) f (0)  f (5)
(d) f (4)  f (1)
23) The household assets in mutual funds measured in billions of dollars, for the years 1990 – 2005
can be modeled by f ( x)  105.095x1.5307 billion dollars, where x is the number of years after
1990. According to this model, what were the assets for the year 2003?
24) The function y  92.239 x 0.1585 gives the percent of voter turnout during presidential election
years, with x representing the number of years after 1950. Determine the percent of voter turnout
in 1996.
Section 4.1
25) Graph each of the following functions and state the transformations that have been applied to the
basic function f ( x)  x 2 .
a) f ( x)  x 2  2
b) f ( x)  ( x  3) 2  5
c) f ( x)   x 2  4
26) Determine whether the graph of each equation is symmetric with respect to the x-axis, the yaxis, the origin, or none of these.
a) y  3x 4  1
b) y  x 3  x
c) y  x  12
27) Determine whether each of the following functions is even, odd, or neither.
a) f ( x)   x 3  4 x
b) f ( x)  x 3  x  3
c)
f ( x)  4 x 2  1
28) Write an equation for each of the following and graph.
a) The graph of y  x 3 is shifted 2 units down and shifted 4 units right.
b) The graph of y  x is reflected across the x-axis and shifted up 3 units.
Section 4.2
29) The composition of functions ( f  g )( x) means…
30) Given f ( x)  3x  4 and g ( x)  2 x  5 , find each of the following, in simplified form.
a) ( f  g )( 2)
b) ( f  g )(1)
f
d)  (2)
g
c) ( fg )( 3)
31) Given f ( x)  x 2  3 and g ( x)  4 x  5 , find each of the following.
a) ( f  g )( x)
b) ( f  g )( x)
c) ( fg )( x)
e) ( f  g )( x)
f) ( g  f )( x)
g)
f
d)  (x)
g
( f  g )( 1)
h) ( g  f )( 2)
Section 4.3
32) Tell whether each function is one-to-one, and if it is, then state the inverse.
a) {(1,4), (2,3), (3,4)}
b) {(1,3),(2,2),(3,4)}
e) f ( x)  3 x  9
f) f ( x) 
1
x2
c) f ( x)  3x 2  1
d) y   x 2
g) f ( x)  9  x 2
h) f ( x)  4 x 3  4
Section 4.4
33) Solve each of the following radical equations and check.
a)
3
4x  5  3 x 2  7
b)
2x  1  x  2
c)
3x  2  2  x
Section 5.1
34) Determine if each of the following functions shows exponential growth or exponential decay.
a) y  2 0.1x
b) y  31.4 x
c) y  4e 5 x
d) y  0.83 x
e) y  0.8 3 x
35) Radioactive lead decays to polonium. If you begin with 1000 grams of radioactive lead, the
amount remaining after t years is given by A(t )  1000e 0.032t .
a) Find the amount remaining after 50 years.
b) Find the half life of radioactive lead, to the nearest year.
36) If a retired couple has a fixed income of $70,000 per year, the purchasing power (real value of
money) after t years of 5% inflation is given by the equation P  70,000(0.95) t dollars.
a) What is the purchasing power of their income after 6 years?
b) When will their purchasing power be $35,000?
37) If you begin with 1000 grams of Carbon-14, the amount remaining after t years is given by
A(t )  1000e 0.00012378t .
a) How much remains after 1500 years?
b) Find the half-life of Carbon-14.
c) When will there be 100 grams remaining?
Sections 5.2, 5.3, & 5.4
38) Write each exponential equation in its equivalent logarithmic form and vice-versa.
a) 2 5  32
b) 25 x  5
c) log 6 36  2
d) log x 30  5
39) Solve each equation below and give the exact values as well as decimal approximations, to 4
decimal places.
a) log 5 x  3
b) log( 5 x)  2
e) ln( x  2)  3
f) 3 log 6 ( x)  5  4
h) 2500  e 5 x
i) 2e 4 x  4  8
d) 12  3 ln x  6
c) log( x  750)  3
g) 10  4 ln x  46
j) 6(2 5 x )  894
k) 1000  700(10 x )
40) Write each expression as a single logarithm with coefficient 1. Simplify your answers as
much as possible. (Assume that all variables represent positive numbers.)
1
a) 2 ln( 3a)  ln b
b) 2 log 4 x 3  log 4 p 3
c) 5 log a x  7 log a y 6
3
41) The population in a certain city was 53,000 in 2000 and its future size is predicted to be
P  53,000e 0.015t people, where t is the number of years after 2000. Determine when the
population will reach 60,000.
42) During a 5 year period of constant inflation, the value of a $100,000 property increases according
to the equation v  100,000e 0.03t dollars. In how many years will the value of this building be
double its current value?
Section 5.5
43) If $9000 is invested for 10 years at 5.5% interest, find the future value that results if the interest
is compounded
a) annually
b) monthly
c) weekly
d) continuously
Section 6.1
44) Determine whether each polynomial graphed below is cubic or quartic, has a positive or negative
leading coefficient, and estimate the x-intercepts. (x-axis scale is 1 unit per tick mark.)
a)
b)
c)
d)
e)
45) The cubic function y  1.832 x 3  22.111x 2  55.367 x  290.659 models the U.S. debt to the
United Nations, in millions of dollars, as a function of the number of years after 1990.
a) According to the model, what was the approximate debt in 1998?
b) Graph the equation on the window [0, 15] by [0 , 400] to find the approximate year when
this debt will be zero.
c) For the given interval, find the year when the local maximum occurs.
Section 6.3
46) Given that the zeros of a polynomial P (x ) are -3, -1, 3 & 9, give a possible equation for P (x ) .
47) Change your equation from #46 so that the end behavior for P (x ) is reversed.
48) The revenue from the sale of a product is given by the function R  12,000 x  0.003x 3 . Use
factoring and the root method to find the number of units that must be sold to give zero revenue.
Verify your results graphically.
49) The price from the sale of a product is given by the function p  100,000  0.1x 2 , where x is the
number of units sold, so the revenue function for the product is R  px  (100,000  0.1x 2 ) x .
Find the number of units that must be sold to give zero revenue. Verify your results graphically.