Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
Solve the following system for x, y, and z. (Answers are given in that order)
x+y+z= 2
x –y+z= – 2
x–y–z= 0
1)
A)
2, –2, –2
B)
4, 2, –4
C)
1, 2, –1
D)
–1, 2, 1
E)
2, 2, –2
2)
Which of the following statements are true?
I. Slope is not defined for a vertical line.
II. A line that falls from left to right has a negative slope.
III. A line with slope 1
3 is more nearly horizontal than a line with slope 2
3.
2)
A)
I only
B)
II only
C)
I and II only
D)
I and III only
E)
all of the above
3)
Find the maximum value of f(x) = 7 – 2x–x2.
3)
A)
7
B)
8
C)
9
D)
10
E)
11
4)
The number of solutions of the system x2+y2= 7
x2–y2= 1 is
4)
A)
zero.
B)
one.
C)
two.
D)
three.
E)
four.
5)
The slope of the line passing through the points (–4, 5) and (3, –2) is
5)
A)
3.
B)
1.
C)
–7.
D)
–1.
E)
–3.
Answer:
6)
Find the x–coordinate of the vertex of a graph of the quadratic function y=f(x) = 4x2– 2x+ 3.
6)
A)
1
4
B)
–1
4
C)
–1
2
D)
1
2
E)
none of the above
Answer:
7)
The demand function for a manufacturer’s product is p=f(q) = 800 – 2q, where p is the price (in
dollars) per unit when q units are demanded (per week). Find the level of production that
maximizes the manufacturer’s total revenue.
7)
A)
125
B)
200
C)
100
D)
175
E)
150
Answer:
Answer:
8)
Suppose q and p are linearly related such that p= 30 when q = 5, and p= 50 when q = 7. Find p
when q= 15.
8)
A)
120
B)
130
C)
140
D)
100
E)
110
9)
The slope and y–intercept of the line 6x– 5y+ 4 = 0 are
9)
A)
6
5 and 4
5, respectively.
B)
5
6 and 5
4, respectively.
C)
6
5 and 4, respectively.
D)
–5
6 and 2
3, respectively.
E)
–5 and 4, respectively.
10)
Suppose f is a linear function with slope 2 and f(1) = 3. Find f(2).
10)
A)
1
B)
2
C)
3
D)
4
E)
5
11)
A manufacturer sells his product at $23 per unit, selling all he produces. His fixed cost is $18,000
and his variable cost per unit is $18.50. The level of production at which the manufacturer breaks
even is
11)
A)
3000 units.
B)
3500 units.
C)
4000 units.
D)
4500 units.
E)
5000 units.
12)
The slope of the line x= 2 is
12)
A)
2.
B)
3.
C)
0.
D)
–2.
E)
not defined.
13)
One solution of the system x –y2= 0
3x+ 2y– 5 = 0 is x= 1 and y= 1. Another solution is
13)
A)
x= 1, y= – 1
B)
x= 0, y= 0
C)
x=9
25 , y= – 3
5
D)
x=25
9, y= – 5
3
E)
x=49
2, y= – 7
2
14)
Suppose f is a linear function with slope –3 and f(2) = 5. Find f(x).
14)
A)
f(x) = – 3x+ 11
B)
f(x) = – 3x+ 2
C)
f(x) = – 3x+ 7
D)
f(x) = – 3x + 13
E)
f(x) = – 3x+ 5
15)
Suppose that a manufacturer will place on the market 80 units of a product when the price is $10
per unit, and 100 units when the price is $12 per unit. Find the supply equation for the product
assuming that price p and quantity q are linearly related.
15)
A)
p= 10q+ 12
B)
p= 10q– 988
C)
p=1
10 q+ 10
D)
p=1
10 q+ 2
E)
p= – 1
10 q+ 12
16)
An equation of the line with slope 8 and y–intercept 5 is
16)
A)
y= 5x+ 8.
B)
y= 8x+ 5.
C)
16x– 2y+ 5 = 0.
D)
5x– 3y+ 8 = 0.
E)
8x+ 2y+ 5 = 0.
17)
The solution of the system
2
3x–1
2y=5
6
2
5x–3
10 y=1
2
is
17)
A)
x= 5, y= – 1
B)
the coordinates of any point on the line y =4
3x–5
3
C)
x= – 1
6, y=7
10
D)
x=4
3, y=3
5
E)
no solution
18)
Suppose f(1) = – 5and f(–2) = 4. Find f(x) if f is a linear function.
18)
A)
f(x) = – x
3+14
3
B)
f(x) =x
3–16
3
C)
f(x) = – 3x+ 5
D)
f(x) =x
3+2
3
E)
f(x) = – 3x– 2
19)
Suppose f(0) = 4 and f(3) = 8. Find f(12) if f is a linear function.
19)
A)
24
B)
–3
C)
20
D)
13
E)
6
20)
An automobile factory produces two models. The first model requires 1 hour to paint and 1
2 hour
to polish. The second requires 1 hour for each process. During each hour that the assembly line is
operating, there are 100 hours available for painting and 80 hours for polishing. How many of the
first model can be produced each hour if all the hours available are to be used?
20)
A)
70
B)
60
C)
50
D)
40
E)
30
21)
A general linear equation of the line that has slope –2 and that passes through the point (1, 3) is
21)
A)
2x+y+ 1 = 0.
B)
2x–y+ 1 = 0.
C)
2x+y– 5 = 0
D)
2x+y– 7 = 0
E)
3x+y+ 5 = 0
22)
A line has slope 3
2 and passes through the point (4, 2). The point on the line that has its second
coordinate equal to –5 is
22)
A)
–1
3, –5 .
B)
–15
2, –5 .
C)
–3
2, –5 .
D)
–2
3, –5 .
E)
–7
2, –5 .
23)
A line passes through the points (1, 1) and (9, 8). The point on it that has a y–coordinate of –6 is
23)
A)
(4, –6).
B)
(–5, –6).
C)
(5, –6).
D)
(–7, –6).
E)
(6, –6).
24)
The slope of the line 4x+ 5y+ 3 = 0 is
24)
A)
4
5.
B)
–4.
C)
–4
3.
D)
4.
E)
–4
5.
25)
Suppose that consumers will demand 100 units of a product when the price is $10 per unit, and 120
units when the price is $8 per unit. Assuming that price p and quantity q are linearly related, find
the price at which 90 units are demanded.
25)
A)
$11
B)
$13
C)
$9
D)
$7
E)
$12
26)
An x–value of a solution of the system x – y – 1 =0
y=x+ 5 is
26)
A)
11.
B)
–1.
C)
1.
D)
4.
E)
–4.
27)
The y–intercept of the line determined by the points (–1, –4) and (–2, 5) is
27)
A)
–13.
B)
–15.
C)
7.
D)
13.
E)
–7.
28)
Find the vertex of the graph of the quadratic function y=f(x) = 4 + 12x– 3x2 .
28)
A)
(2, 4)
B)
(–2, –32)
C)
(–2, 4)
D)
(4, 20)
E)
(2, 16)
29)
If you solve the following system, what is the value of y? 3x+ 2y=26
2x+ 3y=29
29)
A)
3
B)
4
C)
5
D)
6
E)
7
30)
Find the minimum value of g(x) = 3x2– 3x+ 4.
30)
A)
1
2
B)
13
4
C)
4
D)
–5
2
E)
0
31)
If you solve the following system 2x– 3y=1
x + 5y=7, what is the value of y?
31)
A)
1
B)
2
C)
–1
D)
5
E)
–2
32)
The slope of the line 4x– 8y+ 3 = 0 is
32)
A)
2.
B)
–2.
C)
–1
2.
D)
4.
E)
1
2.
33)
The supply and demand equations for a product are p=1
10 q+ 20 and p= 200 –1
2q, respectively,
where q represents the number of units and p represents the price per unit in dollars. The
equilibrium price is
33)
A)
$10.
B)
$20.
C)
$30.
D)
$40.
E)
$50.
34)
The y–intercept of the line 3x+ 5y+ 4 = 0 is
34)
A)
4.
B)
–4.
C)
–4
5.
D)
–4
3.
E)
4
5.
35)
An equation of the straight line passing through (1, –5) and (–2, 4) is
35)
A)
x+ 3y– 14 = 0.
B)
3x+y+ 2 = 0.
C)
x – 3y– 16 = 0.
D)
3x+y– 5 = 0.
E)
x– 3y+ 2 = 0.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
36)
Find a general linear equation of the line that passes through point (1, –2) and has slope 3.
36)
37)
Sketch the graph of y= 3.
37)
38)
Find the range of the function y=f(x) = 4x2– 16x+ 1.
38)
39)
Solve the following system algebraically: 12x– 6y= 7
2x+ 9y=20x+ 3
39)
40)
Find the equilibrium point if the demand equation for a product is p=q
20 – 3 and the
supply equation is p=80
q.
40)
41)
Find the slope of the line passing through the points (3, 9) and (2, –5).
41)
42)
Suppose the cost to produce 100 units of a product is $5000, and the cost to produce 125
units is $6000. If cost c is linearly related to output q, find an equation relating c and q.
42)
43)
Find a general linear equation of the line that passes through the points (–2, 5) and (5, 2).
43)
44)
An automobile factory produces two models. The first model requires 7 widgets and 10
shims. The second model requires 15 widgets and 21 shims. The factory can obtain 800
widgets and 1130 shims per hour. How many cars of each model can it produce per hour if
all parts available are used?
44)
45)
For the parabola y= f(x) =x2– 2x – 8, find: (a) the vertex, (b) the y–intercept, and (c) the
x–intercepts.
45)
46)
Determine a linear function f(x), given f(2) = 0.5; f(1) = – 1.
46)
47)
A botanist collected data from an experiment on the amount of oxygen released by a plant
given different amounts of an experimental fertilizer. He graphs the data with grams of
fertilizer given in a day on the x–axis and milliliters of oxygen produced in a day on the
y–axis. His data points are: (1, 0.02), (3, 0.02), (5, 0.02),
(7, 0.02). Find an equation for the line which fits this data and interpret its meaning.
47)
48)
A pharmacist has two solutions that contain different concentrations of the same
medication. One solution contains a 12% concentration of the medication and the other
contains a 7% concentration. How many cubic centimeters of each should she mix to obtain
40 cc of an 8% concentration?
48)
49)
A California homeowner with a 30–year fixed rate mortgage pays $170,000 after 5 years
and $266,000 after 9 years. Use a graphing calculator to graph the resulting equation and
determine how much the homeowner will have paid when the mortgage is paid off in 30
years.
49)
50)
Solve the system: y=4x–x2
y=x2– 6
50)
51)
Find the slope of the linear function f(t) =3 – 4t
5.
51)
52)
A man standing on a pitcher’s mound throws a ball straight up with an initial velocity of
32 feet per second. The height h of the ball in feet t seconds after it was thrown is described
by the function h(t) = – 16t2+ 32t+ 8. Use a graphing calculator to graph the function.
52)
53)
Two species of monkey, A and B, live in one enclosure at the zoo where they are fed two
vitamin supplements. Each day they receive 350 grams of the first supplement and 500
grams of the second supplement. Each monkey of species A requires 25 g of the first
supplement and 10 g of the second supplement. Each monkey of species B requires 15 g of
the first supplement and 30 g of the second supplement. How many of each species of
monkey will the enclosure support so that all of the supplements are consumed each day?
53)
54)
A manufacturer sells his product at $12.50 per unit, selling all he produces. His fixed cost is
$5,000 and his variable cost per unit is $8.50. (a) At what level of production will he break
even? (b) At what level of production will he have a profit of $10,000?
54)
55)
True or False: The function (2x2 + 3)2 is a Quadratic Function.
55)
56)
Two species of fish, A and B, are raised in one pond at a fish farm where they are fed two
vitamin supplements. Each day they receive 100 grams of the first supplement and 200
grams of the second supplement. Each fish of species A requires 15 mg of the first
supplement and 30 mg of the second supplement. Each fish of species B requires 20 mg of
the first supplement and 40 mg of the second supplement. How many of each species of
fish will the pond support so that all of the supplements are consumed each day? Use a
graphing calculator to graph both of your equations at the same time. What do you notice
about the graphs?
56)
12
57)
Suppose f is a linear function with slope 2 and such that f(–3) = 8. Find f(x).
57)
58)
Find the Break Even Point for a product whose Total Revenue, yTR, (in $) and Total Cost,
yTC, (in $) are as follows:
yTR = (10q– 25)q
yTC = 2000 + 75q
58)
59)
The shape of a decorative awning over a storefront can be described by the function y= f(x)
= 0.06x2+ 0.012x+ 8 where y is the height of the edge of the awning (in feet) above the
sidewalk and x is the distance (in feet) from the center of the store’s doorway. Use a
graphing calculator to graph the function.
59)
60)
Find the equation of a line having slope = 0.5 and x–intercept = 1.8.
60)
61)
Find the equation of a line which is parallel to the line 2x+ 3y– 7 = 0 and passes through
the point (–1, 2).
61)
13
62)
Graph the equation 3x + 4y– 12 = 0.
62)
63)
Find the slope of the line 3x+ 9y– 7 = 0.
63)
64)
For the line y= 7x– 3, find (a) the slope and (b) the y–intercept.
64)
65)
Find the slope of the line 4x– 8y+ 5 = 0.
65)
66)
The shape of a rope bridge stretched across a ravine can be described by the function
y= 0.003x2+ 0.006x+ 50, where y is the height of the bridge (in feet) above the bottom of
the ravine and x is the horizontal distance (in feet) from the center of the ravine. A hiker
traveling at night shines his flashlight across the ravine, and the path of the beam of light
can be described by the function y= 0.096x+ 49.4. Where does the beam of light intersect
the rope bridge? Use a graphing calculator to answer the question by finding the point(s) of
intersection of the two equations.
66)
67)
Determine whether the following lines are parallel, perpendicular or neither.
0.1x– 7y + 81 = 0
2x– 140y+ 9 = 0
67)
68)
Solve the system y= 2x2– 5
x2+y= 4
68)
69)
Suppose the variables q and p are linearly related such that p= 3 when q= 20, and p= 5
when q= 15. Find p when q= 12.
69)
70)
For the parabola y= f(x) = – x2+ 7x – 6, find: (a) the vertex, (b) the y–intercept, and (c) the
x–intercepts.
70)
71)
In testing an experimental diet for beef cows, it was determined that the (average) live
weight w (in kilograms) of a cow was statistically a linear function of the number of days d
after the diet was started, where 0 d
300. The weight of a cow starting the diet was 125
kg and 100 days later it was 245 kg. Determine w as a linear function of d and find the
average weight of a cow when d= 200.
71)
72)
The slope of a certain line is 4. If the x–value of a point on the line increases by 3 units, by
how many units does the y–value increase?
72)
73)
A prediction made by early psychology relating the magnitude of a stimulus x to the
magnitude of a response y is expressed by the equation y=kx2, where k is a constant of the
experiment. In an experiment on brightness of light, k= 1. Find the function’s vertex and
graph the equation.
73)
74)
Suppose that a manufacturer will place 1000 units of a product on the market when the
price is $10 per unit, and 1400 units when the price is $12 per unit. Find the supply
equation for the product assuming the price p and quantity q are linearly related.
74)
16
75)
Consider the restricted quadratic function f (x) =x2 + 4x + 6 on x–2. Determine the
restricted function f–1(x) graphically. Graph both f(x) and f–1(x) on the same xy–plane.
75)
76)
The demand function for a manufacturer’s product is p=f(q) = 6 –q where p is price per
unit when q units are demanded by consumers. Find the level of production that will
maximize the manufacturer’s total revenue and determine this revenue.
76)
77)
Two species of fish, A and B, are raised in one pond at a fish farm where they are fed two
vitamin supplements. Each day they receive 100 grams of the first supplement and 200
grams of the second supplement. Each fish of species A requires 20 mg of the first
supplement and 30 mg of the second supplement. Each fish of species B requires 10 mg of
the first supplement and 40 mg of the second supplement. How many of each species of
fish will the pond support so that all of the supplements are consumed each day? Use
elimination by substitution to solve the systems.
77)
78)
In a test of a diet for pigs, the average live weight w (in kilograms) of a pig was a linear
function of the number d of days after the start of the diet, where 0 d
50. The weight of a
pig at the beginning of a diet was 25 kg and thereafter the pig gained 5 kg every nine days.
(a) Express w as a function of d. (b) Find the weight of a pig 36 days after the beginning of
the diet.
78)
79)
The daily profit for an electronics store from the sale of small televisions is given by P(x) =
–x2+ 40x + 825, where x is the number of small televisions sold. Find the function’s vertex
and intercepts, and graph the function.
79)
80)
For the parabola y= f(x) = 2x2– 4x – 6, find: (a) the vertex, (b) the y–intercept, and (c) the
x–intercepts.
80)
81)
Suppose that consumers will demand 800 units of a product when the price is $10 per unit,
and 1000 units when the price is $8 per unit. Find the demand equation for the product
assuming that price p and quantity q are linearly related.
81)
82)
Graph the general linear form of the child dosage equation whose slope–intercept form is
y=1080
24 t+1080
24 .
82)
83)
The equation of a certain line is 3(x– 4) – (y+ 1) = 4. Find: (a) the slope–intercept form and
(b) a general linear form.
83)
84)
A printer charges a fixed setup cost plus a charge for every copy of single page flyers. If x is
the number of copies requested, the total cost of a printing job can be described by the
function f(x) = 0.02x+ 80. Graph the function on your graphing calculator.
84)
85)
Solve the following system algebraically: 3y– 2x=4
4x– 6y= – 8
85)
86)
A bank charges $10 for a money order. What is an equation for the relationship between
the fee F charged and the amount of the money order?
86)
87)
Determine whether the following lines are parallel, perpendicular or neither.
12x+ 4y= 16
15x+ 5y= 23
87)
88)
The diagram below shows lines with slopes 0, 1, –1, 3, and –3. What is the slope of (a) line
L1 and (b) line L4?
88)
89)
Suppose f is a linear function such that f(0) = 6 and f(3) = 4. Find f(x).
89)