Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1)
For f(t) = 3 – 5t, find f(a + h) – f(a)
h.
1)
Solve the problem.
2)
In the table below, the amount of the U.S. minimum wage is listed for selected years.
U.S. Minimum Wage
Year 1961 1967 1974 1980 1981 1990 1991 1996 1997
Wage $1.15 $1.40 $2.00 $3.10 $3.35 $3.80 $4.25 $4.75 $5.15
Find an exponential regression model of the form y = a ·bx, where y represents the U.S.
minimum wage x years after 1960. Round a and b to four decimal places. According to this
model, what will the minimum wage be in 2005? In 2010?
2)
Provide an appropriate response.
3)
The following graph represents the result of applying a sequence of transformations to the
graph of a basic function. Identify the basic function and describe the transformation(s).
Write the equation for the given graph.
3)
1
4)
Let T be the set of teachers at a high school and let S be the set of students enrolled at that
school. Determine which of the following correspondences define a function. Explain.
(A) A student corresponds to the teacher if the student is enrolled in the teacher’s class.
(B) A student corresponds to every teacher of the school.
4)
5)
Only one of the following functions has domain which is not equal to all real numbers.
State which function and state its domain.
(A) h(x) =4x2– 3x – 5 (B) f(x) =2x
48 – x (C) g(x) =x + 7
2
5)
6)
If f(x) =x – 3 if x < 2
x2 if x 2 , what is the definition of g(x), the function whose graph is
obtained by shifting f(x)’s graph right 5 units and down 1 unit?
6)
2
7)
Graph f(x) = – x2– x + 6 and indicate the maximum or minimum value of f(x), whichever
exists.
7)
8)
Find the vertex and the maximum or minimum of the quadratic function f(x) = – x2– 4x + 5
by first writing f in standard form. State the range of f and find the intercepts of f .
8)
3
Solve the problem.
9)
The financial department of a company that produces digital cameras arrived at the
following price–demand function and the corresponding revenue function:
p(x) = 95.4 – 6x price–demand
R(x) = x · p(x) = x(95.4 – 6x) revenue function
The function p(x) is the wholesale price per camera at which x million cameras can be sold
and R(x) is the corresponding revenue (in million dollars). Both functions have domain 1
x 15. They also found the cost function to be C(x) = 150 + 15.1x (in million dollars) for
manufacturing and selling x cameras. Find the profit function and determine the
approximate number of cameras, rounded to the nearest hundredths, that should be sold
for maximum profit.
9)
Provide an appropriate response.
10)
The following graph represents the result of applying a sequence of transformations to the
graph of a basic function. Identify the basic function and describe the transformation(s).
Write the equation for the given graph.
10)
11)
For f(t) = 3t + 2 and g(t) = 2 –t2, find 4f(3) – g(–3) + g(0).
11)
12)
If g(x) = – 4x2+ x – 9, find g(–2), g(1), and g 3
2.
12)
4
Use the REGRESSION feature on a graphing calculator.
13)
A particular bacterium is found to have a doubling time of 20 minutes. If a laboratory
culture begins with a population of 300 of this bacteria and there is no change in the
growth rate, how many bacteria will be present in 55 minutes? Use six decimal places in
the interim calculation for the growth rate.
13)
Solve the problem.
14)
The financial department of a company that manufactures portable MP3 players arrived at
the following daily cost equation for manufacturing x MP3 players per day:
C(x) = 1500 + 105x +x2. The average cost per unit at a production level of players per day
is C(x) =C(x)
x.
(A) Find the rational function C.
(B) Graph the average cost function on a graphing utility for 10 x 200.
(C) Use the appropriate command on a graphing utility to find the daily production level
(to the nearest integer) at which the average cost per player is a minimum. What is the
minimum average cost (to the nearest cent)?
14)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
15)
15)
A)
$425.00
B)
$467.03
C)
$7083.33
D)
$16.67
For the rational function below (i) Find the intercepts for the graph; (ii) Determine the domain; (iii) Find any vertical or
horizontal asymptotes for the graph; (iv) Sketch any asymptotes as dashed lines. Then sketch the graph of y = f(x).
16)
16)
A)
(i) x intercept: 3; y intercept: 3
4
(ii) Domain: all real numbers except 4
(iii) Vertical asymptote: x = 4; horizontal asymptote: y = 1
(iv)
B)
(i) x intercept: –5; y intercept: 3
4
(ii) Domain: all real numbers except –4
(iii) Vertical asymptote: x = – 4; horizontal asymptote: y = 1
(iv)
6
C)
(i) x intercept: –3; y intercept: 3
4
(ii) Domain: all real numbers except –4
(iii) Vertical asymptote: x = – 4; horizontal asymptote: y = 1
(iv)
D)
(i) x intercept: 5; y intercept: 3
4
(ii) Domain: all real numbers except 4
(iii) Vertical asymptote: x = 4; horizontal asymptote: y = 1
(iv)
Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x
to which there corresponds more than one value of y.
17)
17)
A)
A function with domain
B)
Not a function; for example, when x = 10, y = ±1
18)
18)
A)
A function with domain all real numbers except x = – 3
B)
Not a function; for example, when x =4, y = ±3
Convert to a logarithmic equation.
19)
19)
A)
2 =log25 5
B)
2 =log 5 25
C)
25 =log5 2
D)
5 =log2 25
The graph that follows is the graph of a polynomial function. (i) What is the minimum degree of a polynomial function
that could have the graph? (ii) Is the leading coefficient of the polynomial negative or positive?
20)
20)
A)
(i) 4
(ii) Positive
B)
(i) 3
(ii) Negative
C)
(i) 3
(ii) Positive
D)
(i) 4
(ii) Negative
Use the REGRESSION feature on a graphing calculator.
21)
21)
A)
y =0.0903x
B)
y =1.3384x
C)
y = 0.0903 ·1.3384x
D)
y = 1.3384 ·0.0903x
Solve the problem.
22)
22)
A)
$2281.25
B)
$2844.31
C)
$1594.31
D)
$1031.25
Provide an appropriate response.
23)
23)
A)
8 because this is the degree of the polynomial.
B)
0 because a polynomial of even degree may not cross the x axis at all.
C)
1 because a polynomial of even degree crosses the x axis at least once.
D)
Not enough information is given.
For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept.
24)
24)
A)
(i) 1
(ii) –5
4
(iii) 5
B)
(i) 1
(ii) 5
(iii) 5
4
C)
(i) 1
(ii) – 4
5
(iii) 4
D)
(i) 1
(ii) 5
4
(iii) 5
Determine whether the graph is the graph of a function.
25)
25)
A)
function
B)
not a function
Use a calculator to evaluate the expression. Round the result to five decimal places.
26)
26)
A)
1.73388
B)
1.56585
C)
3.60550
D)
0.57674
Determine whether the relation represents a function. If it is a function, state the domain and range.
27)
27)
A)
function
domain: {–1, 2, -2, 14}
range: {-3, -2, 0, 2, 4}
B)
function
domain: {-3, -2, 0, 2, 4}
range: {–1, 2, -2, 14}
C)
not a function
28)
28)
A)
function
domain: {Bob, Ann, Dave}
range: {carrots, peas, squash}
B)
function
domain: {carrots, peas, squash}
range: {Bob, Ann, Dave}
C)
not a function
Solve the problem.
29)
29)
A)
605 grams
B)
9 grams
C)
800 grams
D)
1 gram
Write an equation for the lowest–degree polynomial function with the graph and intercepts shown in the figure.
30)
30)
A)
f(x) =x2+ 40x – 13
B)
f(x) =x2+ 40x + 13
C)
f(x) =x2– 13x + 40
D)
f(x) =x2+ 13x + 40
Use a calculator to evaluate the expression. Round the result to five decimal places.
31)
31)
A)
51.237
B)
3.93646
C)
1.70958
D)
Undefined
For the given function, find each of the following:
(A) Intercepts
(B) Vertex
(C) Maximum or minimum
(D) Range
32)
32)
A)
(A) x–intercepts: – 5, -1; y–intercept: -5
(B) Vertex (-3, 4)
(C) Maximum: 4
(D) y 4
B)
(A) x–intercepts: 1, 5; y–intercept: -5
(B) Vertex (-3, 4)
(C) Maximum: 4
(D) y 4
C)
(A) x–intercepts: – 5, -1; y–intercept: -5
(B) Vertex (-3, 4)
(C) Minimum: 4
(D) y 4
D)
(A) x–intercepts: – 5, -1; y–intercept: -5
(B) Vertex (3, -4)
(C) Maximum: 4
(D) y 4
Find the function value.
33)
33)
A)
2
9
B)
13
72
C)
13
66
D)
13
64
Give the domain and range of the function.
34)
34)
A)
Domain: all real numbers; Range: all real numbers
B)
Domain: [– 6, ); Range: all real numbers
C)
Domain: all real numbers; Range: [0, )
D)
Domain: all real numbers; Range: [– 6, )
For the following problem, (i) graph f and g in the same coordinate system; (ii) solve f(x) = g(x) algebraically to two
decimal places; (iii) solve f(x) > g(x) using parts i and ii; (iv) solve f(x) < g(x) using parts i and ii.
13
35)
35)
A)
(i) f is the curve, g is the line
(ii) 0.58, 7.98
(iii) 0.58 < x < 7.98
(iv) 0 x < 0.58 or 7.98 < x 8
B)
(i) f is the curve, g is the line
(ii) 0.58, 6.92
(iii) 0.58 < x < 6.92
(iv) 0 x < 0.58 or 6.92 < x 8
14
C)
(i) f is the curve, g is the line
(ii) 0.61, 7.02
(iii) 0.61 < x < 7.02
(iv) 0 x < 0.61 or 7.02 < x 8
D)
(i) f is the curve, g is the line
(ii) 0.61, 7.98
(iii) 0.61 < x < 7.98
(iv) 0 x < 0.61 or 7.98 < x 8
Solve the problem.
36)
A retail chain sells washing machines. The retail price p(x) (in dollars) and the weekly demand x
for a particular model are related by the function p(x) = 625 – 5 x, where 50 x 500. (i) Describe
how the graph of the function p can be obtained from the graph of one of the six basic functions: y
= x, y =x2,y =x3, y =x, y =3x, or y =x. (ii) Sketch a graph of function p using part (i) as an
aid.
36)
15
A)
(i) The graph of the basic function y =x is
reflected in the x–axis and vertically
expanded by a factor of 5.
(ii)
B)
(i) The graph of the basic function y =x is
vertically expanded by a factor of 5,
and shifted up 625 units.
(ii)
16
C)
(i) The graph of the basic function y =x is
vertically expanded by a factor of 625,
and shifted up 5 units.
(ii)
D)
(i) The graph of the basic function y =x is
reflected in the x–axis, vertically expanded
by a factor of 5, and shifted up 625 units.
(ii)
The graph that follows is the graph of a polynomial function. (i) What is the minimum degree of a polynomial function
that could have the graph? (ii) Is the leading coefficient of the polynomial negative or positive?
37)
37)
A)
(i) 2
(ii) Negative
B)
(i) 3
(ii) Negative
C)
(i) 2
(ii) Positive
D)
(i) 3
(ii) Positive
Solve the problem.
38)
38)
A)
3.9 million
B)
4.8 million
C)
8.0 million
D)
7.8 million
Determine whether there is a maximum or minimum value for the given function, and find that value.
39)
39)
A)
Minimum: 0
B)
Maximum: –4
C)
Minimum: 4
D)
Maximum: 10
18
The graph that follows is the graph of a polynomial function. (i) What is the minimum degree of a polynomial function
that could have the graph? (ii) Is the leading coefficient of the polynomial negative or positive?
40)
40)
A)
(i) 2
(ii) Negative
B)
(i) 3
(ii) Negative
C)
(i) 3
(ii) Positive
D)
(i) 2
(ii) Positive
Convert to a logarithmic equation.
41)
41)
A)
log 7 t = e
B)
log 7 e = t
C)
ln 7 = t
D)
ln t = 7
Determine whether the relation represents a function. If it is a function, state the domain and range.
42)
42)
A)
function
domain: {5, 9, 13, 17}
range: {10, 18, 26, 34}
B)
function
domain:{10, 18, 26, 34}
range: {5, 9, 13, 17}
C)
not a function
Provide an appropriate response.
43)
43)
A)
11 because this is the degree of the polynomial.
B)
1 because a polynomial of odd degree crosses the x axis at least once.
C)
0 because a polynomial of odd degree may not cross the x axis at all.
D)
Not enough information is given.
Use the properties of logarithms to solve.
44)
44)
A)
10
B)
–10, 1
C)
–10
D)
–1, 10
A
Compute and simplify the difference quotient f(x +
h) – f(x)
h, h 0.
45)
45)
A)
10x2+ 5h+ 7x
B)
15x – 7h + 14
C)
10x + 5h + 7
D)
10x + 7
C
Solve the problem.
46)
46)
A)
8 atm
B)
7 atm
C)
200 atm
D)
201 atm
B
B