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Ch. 3 Lines, Parabolas, and Systems
3.1 Lines
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1) Find the slope of the line passing through the points (5, -3) and (2, -1).
2) Find the slope of the line passing through the points (3, 9) and (2, -5).
3) The slope of the line passing through the points (4, 9) and (6, k) is 5. Find k.
4) For the line y = 7x - 3, find (a) the slope and (b) the y-intercept.
5) Find the slope of the line 4x - 8y + 5 = 0.
6) Find the slope of the line 3x + 9y - 7 = 0.
7) What is the slope of a horizontal line?
8) 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 ?
9) 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?
10) Graph the equation 3x + 4y - 12 = 0.
y
10
5
-10
-5
5
10
x
-5
-10
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11) Graph the equation 5x + y + 8 = 0.
y
10
5
-10
-5
5
10
x
10
x
10
x
-5
-10
12) Sketch the graph of x = 4.
y
10
5
-10
-5
5
-5
-10
13) Sketch the graph of y = 3.
y
10
5
-10
-5
5
-5
-10
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14) Sketch the graph of -3(x - 6) - 7(y + 9) = 0.
y
10
5
-10
-5
5
10
x
-5
-10
15) For the straight line 2x + y - 3 = 0 find: (a) the slope; (b) the y-intercept; and
(c) sketch the graph.
y
10
5
-10
-5
5
10
x
-5
-10
16) The equation of a certain line is y + 2 = -2(x - 3). Find: (a) the slope-intercept form and (b) a general linear
form.
17) 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.
18) Find the equation of the line with y-intercept 4 and slope -
2
.
3
19) Find an equation of the line that passes through the origin and that has slope -5.
20) Find the slope-intercept form of an equation of the line that passes through the point (2, 0) and has slope 4.
21) Find a general linear equation of the line that passes through point (1, -2) and has slope 3.
22) Find a general linear equation of the line that passes through point (-6, 4) and has slope -2.
23) Find a general linear equation of the line that passes through the points (-2, 5) and (5, 2).
24) Find a general linear equation of the line that passes through the points (4, -3) and (6, -7).
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25) Determine an equation of the vertical line that passes through the point (3, -6).
26) Find an equation of the horizontal line that passes through the point (5, 6).
27) Find the slope-intercept form of the line that passes through (0,4) and (1,1).
28) Find the slope-intercept form of the line that passes through (2,-3) and (-4,7).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
29) The slope of the line passing through the points (-4, 5) and (3, -2) is
A) 1.
B) -1.
C) 3.
D) -3.
E) -7.
C) -2.
D) 3.
E) not defined.
30) The slope of the line x = 2 is
A) 0.
B) 2.
31) 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.
2
1
is more nearly horizontal than a line with slope .
III. A line with slope
3
3
A) I only
B) II only
C) I and II only
D) I and III only
E) all of the above
32) The slope of the line 4x + 5y + 3 = 0 is
A) 4.
B) -4.
C)
4
.
5
D) -
4
.
5
E) -
C)
1
.
2
D) -
1
.
2
E) 4.
C)
4
.
5
D) -
4
.
5
E) -
4
.
3
33) The slope of the line 4x - 8y + 3 = 0 is
A) 2.
B) -2.
34) The y-intercept of the line 3x + 5y + 4 = 0 is
A) 4.
B) -4.
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4
.
3
35) The slope and y-intercept of the line 6x - 5y + 4 = 0 are
6
A) and 4, respectively.
5
B) -5 and 4, respectively.
4
6
C) and , respectively.
5
5
D) E)
2
5
and , respectively.
3
6
5
5
and , respectively.
4
6
36) An equation of the line with slope 8 and y-intercept 5 is
A) y = 8x + 5.
B) y = 5x + 8.
C) 8x + 2y + 5 = 0.
D) 5x - 3y + 8 = 0.
E) 16x - 2y + 5 = 0.
37) A general linear equation of the line that has slope -2 and that passes through the point (1, 3) is
A) 2x + y + 1 = 0.
B) 2x - y + 1 = 0.
C) 2x + y - 7 = 0
D) 3x + y + 5 = 0
E) 2x + y - 5 = 0
38) An equation of the straight line passing through (1, -5) and (-2, 4) is
A) x - 3y + 2 = 0.
B) x - 3y - 16 = 0.
C) x + 3y - 14 = 0.
D) 3x + y - 5 = 0.
E) 3x + y + 2 = 0.
39) The y-intercept of the line determined by the points (-1, -4) and (-2, 5) is
A) 7.
B) -7.
C) 13.
D) -13.
E) -15.
40) A line passes through the points (1, 1) and (9, 8). The point on it that has a y-coordinate of -6 is
A) (-7, -6).
41) A line has slope
equal to -5 is
3
A) - , -5 .
2
B) (6, -6).
C) (-5, -6).
D) (4, -6).
E) (5, -6).
3
and passes through the point (4, 2). The point on the line that has its second coordinate
2
B) -
1
, -5 .
3
C) -
2
, -5 .
3
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D) -
15
, -5 .
2
E) -
7
, -5 .
2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
42) Determine whether the following lines are parallel, perpendicular or neither.
3x - 2y = -17
2x + 3y = -1
43) Determine whether the following lines are parallel, perpendicular or neither.
3x - 6y = 19
18x + 9y = -9
44) Determine whether the following lines are parallel, perpendicular or neither.
9x + 3y = 12
6x + 2y = 9
45) Determine whether the following lines are parallel, perpendicular or neither.
9x + 3y = 12
18x + 6y = 27
46) Determine whether the following lines are parallel, perpendicular or neither.
3x - 12y = 11
64x + 12y = 3
47) Determine whether the following lines are parallel, perpendicular or neither.
0.1x - 7y + 81 = 0
2x - 140y + 9 = 0
48) Determine whether the following lines are parallel, perpendicular or neither.
0.21x - 0.35y + 8 = 0
x + 0.6y - 9 = 0
49) Find the equation of a line which is parallel to the line 2x + 3y - 7 = 0 and passes through the point (-1, 2).
50) Find the equation of a line which is perpendicular to the line 2x + 3y - 57 = 0 and passes through the point
(1, -1).
51) Find the equation of a line which is parallel to the line x = 0.521 and passes through the point (0.1, -7).
52) Find the equation of a line which is perpendicular to the line x = 0.521 and passes through the point (0.1, -7).
53) Find the equation of a line having slope = 1 and x-intercept = -2.
54) Find the equation of a line having slope = 0.5 and x-intercept = 1.8.
55) Find the x-intercept made by the line that passes through the points (1, -1) and (-1, 0).
56) Determine the equation of the line which is perpendicular to 2x - y + 3 = 0 and has y-intercept 6.
57) Find the equation of a line having slope 0.5 and x-intercept = 1.3.
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58) Find the slope and y-intercept of a line 2y + 3(x - 1.9) = 0.
59) 10 square yards of a good quality wool carpet costs $400 and 20 square yards costs $800. Use a graphing
calculator to show the relationship between cost and amount purchased. Find and interpret the slope.
60) In 1986 the stock in a biotechnology company traded for $30 per share. In 1996 the company started having
trouble, and the stock price dropped to $10 per share. Use a graphing calculator to show the relationship
between price per share and the year in which it traded. Find and interpret the slope.
61) The average weight of newborn blue whales is 3 tons. By the time they are 7 months old the average weight of
these whales is 23 tons. Draw a line showing the relationship between weight (in tons) and age (in months) of
blue whales. Find and interpret the slope.
62) The price of computer technology has been dropping steadily for the past ten years. The price of a certain
desktop PC has decreased by $650 per year during this time period. If this PC sold for $6770 ten years ago,
what equation describes the cost C of the PC over the last ten years? Graph your equation on a graphing
calculator. Can you predict the price of a desktop PC a few years into the future (T > 0) with this model?
63) A baby weighs 9 pounds at birth and 30 pounds at age 3. Use a graphing calculator to graph the resulting
equation and determine how much the child will weight at age 12.
64) 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.
65) The stock price of a company has risen at the rate of $5.00 per month over the last year. On January 1 it was
$75.00. Write an equation that shows this relationship.
66) The area of rain forest in a South American country has decreased by 500 acres per year for the last 10 years. It
had 12,000 acres 10 years ago. Use a graphing calculator to graph the resulting equation and determine when
all of the rain forest in this country will be destroyed.
67) A delicatessen owner starts her business with debts of $100,000. After operating for 5 years she has average
profits of $40,000 per year. Use a graphing calculator to graph the resulting equation and determine when the
business will have accumulated $300,000 in profit.
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68) A deep sea diver has spent a week in a submerged research facility at 2000 feet below sea level. He is now
ready to move to a deeper facility at 3500 feet below sea level. He descends at the rate of 50 feet per minute.
Write an equation that shows this relationship, and determine when the diver will reach 3500 feet below sea
level.
69) The relationship between temperature on the Fahrenheit scale and the temperature on the Celsius scale is
5
C = (F - 32). Find the slope and y-intercept of the equation.
9
70) The size of a human fetus more than 12 weeks old can be estimated by the formula L = 1.53t - 6.7, where L is in
centimeters and t is in weeks. An obstetrician uses the length of a fetus, measured in an ultrasound, to
determine the approximate age of the fetus and establish a due date for the mother. The formula must be
rewritten to result in an age t, given a fetal length L. Use a graphing calculator to graph the resulting equation
and verify the y-intercept.
71) A mathematical model can approximate the winning distance for the Olympic discus throw by the formula d =
175 + 1.75t, where d is in feet and t = 0 corresponds to the year 1948. We might want to predict in what year a
certain distance will be exceeded, so rewrite the equation to solve for t. Use a graphing calculator to graph the
resulting equation and find the coordinates of any two points on the line and use them to estimate the slope.
Compare this with your answer.
72) An orthodontist charges $3000 for the phase of treatment for a nine -year old which will provide better
alignment and more room for permanent teeth in the future. What is an equation for the relationship between
cost C of the treatment and the number of visits required to attain the desired result?
73) 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?
74) A coordinate map of a zoo shows the outdoor aviary at (2, 4) and the snack bar at (2, 8). What is an equation for
the line between these two locations?
75) 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.
76) Graph the general linear form of the Fahrenheit-Celsius conversion equation whose slope-intercept form is
9
F = C + 32.
5
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77) Graph the general linear form of the Olympic discus throw equation whose slope -intercept form is
d = 1.75t + 175.
78) Graph the general linear form of the child dosage equation whose slope -intercept form is y =
1080
1080
t+
.
24
24
79) Graph the general linear form of the equation for the size of a human fetus more than 12 weeks old whose
slope-intercept form is L = 1.53t - 6.7.
80) Show that the points A(0, 0), B(0, 3), C(8, 5), and D(12, 3) are the vertices of a trapezoid. (A trapezoid is a
four-sided figure with exactly two sides parallel.)
81) Show that the points A(-1, 2), B(3, -2), and C(-6, -3), are the vertices of a right triangle.
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3.2 Applications and Linear Functions
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
82
1) For the linear function f(x) = -5x + 5, find: (a) the slope and (b) the vertical axis intercept. (c) Sketch the graph of
f.
y
10
5
-10
-5
5
10
x
-5
-10
83
2) For the linear function f(x) = 2x + 1, find: (a) the slope and (b) the vertical axis intercept. (c) Sketch the graph of
f.
y
10
5
-10
-5
5
10
x
-5
-10
3 - 4t
.
5
84
3) Find the slope of the linear function f(t) =
85
4) Suppose f is a linear function such that f(-2) = 5 and f(5) = 2. Find f(x).
86
5) Suppose f is a linear function such that f(0) = 6 and f(3) = 4. Find f(x).
87
6) Suppose f is a linear function with slope 5 and such that f(1) = 4. Find f(x).
88
7) Suppose f is a linear function with slope 2 and such that f(-3) = 8. Find f(x).
89
8) 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.
90
9) 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.
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91
10) 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.
92
11) 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.
93
94
12) 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.
13) When the temperature T (in degrees Celsius) of a certain laboratory animal is reduced, its heart rate r (in beats
per minute) decreases. At a temperature of 37°C, the animal had a heart rate of 200, and at a temperature of
32°C its heart rate was 140. If r is a linear function of T for 26 ≤ T ≤ 38, (a) determine this function and (b)
determine the heart rate at a temperature of 30°C.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
95
14) Suppose f is a linear function with slope -3 and f(2) = 5. Find f(x).
A) f(x) = -3x + 2
B) f(x) = -3x + 5
C) f(x) = -3x + 7
D) f(x) = -3x + 11
E) f(x) = -3x + 13
96
15) Suppose f(1) = -5 and f(-2) = 4. Find f(x) if f is a linear function.
x 16
A) f(x) = 3
3
B) f(x) = -
x 14
+
3
3
C) f(x) = -3x - 2
D) f(x) = -3x + 5
x 2
E) f(x) =
+
3 3
97
16) Suppose f is a linear function with slope 2 and f(1) = 3. Find f(2).
A) 1
98
C) 3
D) 4
E) 5
D) 20
E) 24
17) Suppose f(0) = 4 and f(3) = 8. Find f(12) if f is a linear function.
A) -3
99
B) 2
B) 6
C) 13
18) 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.
A) 100
B) 110
C) 120
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D) 130
E) 140
100
19) 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.
1
q+2
A) p =
10
B) p =
1
q + 10
10
C) p = -
1
q + 12
10
D) p = 10q + 12
E) p = 10q - 988
101
20) 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.
A) $7
B) $9
C) $11
D) $12
E) $13
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
3p - 2
. Find the slope and the y-intercept.
5
102
21) Suppose f(p) is a linear function given by f(p) =
103
22) Suppose that f(x) is a linear function with slope = -3 and y-intercept
104
23) Suppose consumers will demand 30 units of a product when the price is $12 per unit and 22 units when the
price is $16 each. Find the demand equation assuming that it is linear.
105
24) Determine the linear function f(t) with slope = -1 and f(2) = 1.
106
25) Determine a linear function f(x), given f(2) = 0.5; f(1) = -1.
1
, then find the function f(x).
5
107
26) Tickets to an opera at the Masonic Auditorium cost $14 for main floor seats and $10 for the balcony seats. If
$8600 must be collected to meet expenses, what is an equation for the possible combinations of ticket sales to
cover costs?
108
27) An accountant can complete a simple tax return in about 1.5 hours and a complicated return in about 4 hours.
If she works 8 hours per day, find an equation that shows the possible ways that both types of returns can be
completed in a 5-day work week.
109
28) An investor has $12,000 to purchase stock in two companies. If the first stock sells for $45 per share, and the
second stock sells for $62 per share, find an equation that shows the possible ways to purchase the stock.
110
29) The demand per week for a new automobile is 400 units when the price is $16,700 each, and 500 units when the
price is $14,900 each. Find the demand equation for the cars, assuming that it is linear.
111
30) A function describing the value of a house purchased for $180,000 after x years of appreciation is estimated to
be f(x) = 12,000x + 180,000. Graph the function on your graphing calculator.
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112
31) 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.
113
32) The value of an antique figurine is expected to appreciate each year after it is purchased for $550. If x is the
number of years that have passed, the current value of the figurine can be estimated by the function
f(x) = 75x + 550. Graph the function by finding and plotting two points.
114
33) 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.
115
116
117
34) In testing an experimental diet for goats, it was determined that the (average) live weight w (in kilograms) of a
goat was statistically a linear function of the number of days d after the diet was started where 0 ≤ d ≤ 100. The
weight of a goat starting the diet was 12 kg and 25 days later it was 20 kg. Determine w as a linear function of d
and find the average weight of a goat when d = 80.
35) In testing an experimental diet for sheep, it was determined that the (average) live weight w (in kilograms) of a
sheep was statistically a linear function of the number of days d after the diet was started, where 0 ≤ d ≤ 150.
The weight of a sheep starting the diet was 15 kg and 40 days later it was 43 kg. Determine w as a linear
function of d and find the average weight of a sheep when d =100.
36) In testing an experimental diet for horses, it was determined that the (average) live weight w (in kilograms) of a
horse was statistically a linear function of the number of days d after the diet was started where 0 ≤ d ≤ 300. The
weight of a horse starting the diet was 170 kg and 120 days later it was 362 kg. Determine w as a linear function
of d and find the average weight of a horse when d = 250.
3.3 Quadratic Functions
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
118
119
120
1) For the parabola y = f(x) = x 2 - 2x - 8, find: (a) the vertex, (b) the y-intercept, and (c) the x-intercepts.
2) For the parabola y = f(x) = 2x 2 - 4x - 6, find: (a) the vertex, (b) the y-intercept, and (c) the x-intercepts.
3) For the parabola y = f(x) = -x 2 + 7x - 6, find: (a) the vertex, (b) the y-intercept, and (c) the x-intercepts.
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121
122
4) For the parabola y = f(x) = 4 - x - 3x 2 , find: (a) the vertex, (b) the y-intercept, and (c) the x-intercepts.
5) Graph the function y = f(x) = x 2 - 6x and indicate the coordinates of the vertex and intercepts.
y
10
5
-10
-5
5
10
x
-5
-10
123
6) Graph the function y = f(x) = x 2 - 6x + 5 and indicate the coordinates of the vertex and intercepts.
y
10
5
-10
-5
5
10
x
-5
-10
124
7) Graph the function y = f(x) = 2x 2 + 2x - 12 and indicate the coordinates of the vertex and intercepts.
y
10
5
125
-10
-5
5
10
x
-5
-10
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126
8) Graph the function y = f(x) = -x 2 + 5 - 4 and indicate the coordinates of the vertex and intercepts.
y
10
5
-10
-5
5
10
x
-5
-10
127
9) Graph the function y = f(x) = 3 - 2x - x 2 and indicate the coordinates of the vertex and intercepts.
y
10
5
-10
-5
5
10
x
-5
-10
128
10) State whether f(x) = 12x 2 - 24x + 10 has maximum or minimum value and find that value.
129
11) State whether f(x) = 10 + 16x - 4x 2 has maximum or minimum value and find that value.
130
12) Find the range of the function y = f(x) = -x 2 + 3x + 2.
131
13) Find the range of the function y = f(x) = 4x 2 - 16x + 1.
132
14) The demand function for a manufacturerʹs product is p = f(q) = 600 - 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 manufacturerʹs
total revenue and determine this revenue.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
133
15) Find the x-coordinate of the vertex of a graph of the quadratic function y = f(x) = 4x 2 - 2x + 3.
1
A) 4
B)
1
2
C)
1
4
D) -
1
2
E) none of the above
134
16) Find the vertex of the graph of the quadratic function y = f(x) = 4 + 12x - 3x 2 .
A) (4, 20)
135
136
C) (-2, 4)
17) Find the minimum value of g(x) = 3x 2 - 3x + 4.
13
B) 4
C) 0
A)
4
D) (2, 4)
D) -
5
2
E) (2, 16)
E)
1
2
18) Find the maximum value of f(x) = 7 - 2x - x 2 .
A) 7
137
B) (-2, -32)
B) 8
C) 9
D) 10
E) 11
19) The demand function for a manufacturerʹs 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 manufacturerʹs
total revenue.
A) 100
B) 125
C) 150
D) 175
E) 200
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
138
20) True or False: The function (2x 2 + 3)2 is a Quadratic Function.
139
21) Find the vertex and axis of symmetry of the parabola y = 2x 2 + 3x - 5.
140
22) Find the vertex and axis of symmetry of the parabola y = -x 2 + 6x + 3. Also find if it opens upward or
downward.
141
23) State whether the following function has a maximum value or minimum value and find it: f(x) = 2 + 10x - x 2
142
24) Suppose that the vertex of the parabola y = 3x 2 - 6x + k is (1, 2); find k.
143
25) The demand function for a manufacturerʹs 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 manufacturerʹs total revenue and
determine this revenue.
144
26) The daily profit for a lighting store from the sale of a table lamp is given by P(x) = -x 2 + 34x + 35, where x is the
number of table lamps sold. Use a graphing calculator to graph the function.
Page 135
145
27) The daily profit for an electronics store from the sale of small televisions is given by P(x) = -x 2 + 40x + 825,
where x is the number of small televisions sold. Find the functionʹs vertex and intercepts, and graph the
function.
y
x
146
147
28) The daily profit for the garden department of a store from the sale of trees is given by P(x) = -x 2 + 18x + 144,
where x is the number of trees sold. Use a graphing calculator to graph the function.
29) 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 = kx 2 , where k is a constant of the experiment. In an experiment on noise
levels, k = 4. Use a graphing calculator to graph the equation.
148
30) 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 = kx 2 , where k is a constant of the experiment. In an experiment on brightness
of light, k = 1. Find the functionʹs vertex and graph the equation.
y
x
149
31) 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 = kx 2 , where k is a constant of the experiment. In an experiment on odor
intensity, k = 5. Use a graphing calculator to graph the equation.
150
32) A man standing on a pitcherʹs 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.
151
33) A toy rocket is launched straight up from the roof of a garage with an initial velocity of 80 feet per second. The
height h of the rocket in feet t seconds after it was thrown is described by the function h(t) = -16t2 + 80t + 16.
Use a graphing calculator to graph the function.
Page 136
152
34) The shape of a paper streamer suspended above a dance floor can be described by the function
y = f(x) = 0.01x 2 + 0.01x + 7, where y is the height of the streamer (in feet) above the floor and x is the horizontal
distance (in feet) from the center of the room. Use a graphing calculator to graph the function.
153
35) The shape of a decorative awning over a storefront can be described by the function y = f(x) = 0.06x 2 + 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 storeʹs doorway. Use a graphing calculator to graph the function.
154
36) The demand function for an appliance companyʹs line of washing machines is p = 300 - 5q, where p is the price
(in dollars) per unit when q units are demanded (per week) by consumers. Find the level of production that will
maximize the manufacturerʹs total revenue, and determine this revenue.
155
37) Consider the restricted quadratic function f (x) = x 2 + 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.
3.4 Systems of Linear Equations
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
156
1) Solve the following system algebraically:
2x - y = 1
-x + 2y = 7
157
2) Solve the following system algebraically:
5u + v = -2
20u +2v = 1
158
3) Solve the following system algebraically:
3x - 4y = 18
2x + 5y = -11
159
4) Solve the following system algebraically:
5x + 2y = 36
8x - 3y = -54
5) Solve the following system algebraically:
3x + 5y = - 6
2x - 6 = 5y
160
161
6)
1
1
1
x- y=
2
6
4
Solve the following system algebraically:
x+
2
1
y =
3
2
162
7) Solve the following system algebraically:
12x - 6y = 7
2x + 9y = 20x + 3
163
8) Solve the following system algebraically:
8x - 4y = 7
y = 2x - 4
164
9) Solve the following system algebraically:
3y - 2x = 4
4x - 6y = -8
Page 137
165
10) Solve the following system algebraically:
2x + y + z = 0
4x + 3y + 2z = 2
2x - y - 3z = 0
166
11)
167
168
169
Solve the following system algebraically:
2x - y + 3z = 12
x + y - z = -3
x + 2y - 3z = -10
Solve the following system algebraically:
x - z = 14
y + z = 21
x -y + z = -10
12)
13) A manufacturer produces two products, A and B. For each unit of A sold the profit is $8. For each unit of B sold
the profit is $11. From past experience it has been found that 25 percent more of A can be sold than of B. Next
year the manufacturer desires a total profit of $42,000. How many units of each product must be sold?
14) 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?
170
15) How much of each of a 25% (by volume) chemical solution and a 32% solution must be combined to make 75
cubic centimeters of a 28% solution?
171
16) A coffee wholesaler blends together three types of coffee that sell for $1.95, $2.10, and $2.25 per pound so as to
obtain 100 pounds of coffee worth $2.13 per pound. If the wholesaler uses the same amount of the two
higher-priced coffees, how much of each type must be used in the blend?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
172
17) If you solve the following system, what is the value of y?
A) 3
173
B) 4
18) If you solve the following system
A) 2
B) -2
3x + 2y = 26
2x + 3y = 29
C) 5
D) 6
E) 7
D) -1
E) 5
2x - 3y = 1 , what is the value of y?
x + 5y = 7
C) 1
Page 138
174
19)
2
1
5
x- y=
3
2
6
The solution of the system
2
3
1
xy=
5
10
2
is
A) x = 5, y = -1
3
4
B) x = , y =
5
3
C) x = -
1
7
,y=
6
10
D) the coordinates of any point on the line y =
5
4
x3
3
E) no solution
175
20)
Solve the following system for x, y, and z. (Answers are given in that order)
A) 4, 2, -4
176
B) 2, -2, -2
C) 2, 2, -2
x+y+z=2
x - y + z = -2
x-y-z=0
D) 1, 2, -1
21) An automobile factory produces two models. The first model requires 1 hour to paint and
E) -1, 2, 1
1
hour to polish.
2
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?
A) 70
B) 60
C) 50
D) 40
E) 30
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
177
22) By sketching the graphs (using a graphing calculator or graphing paper) solve the system of
2x + 3y = 26
equations.
3x - 2y = 13
178
23) Solve the following system of equations algebraically. (If the system does not have a solution, then say so or if it
has more than one unique solution, then please describe the solutions.)
4x - 10y = 6
5
2
x- y=7
3
3
179
24) Solve the following system of equations algebraically. (If the system does not have a solution, then say so or if it
has more than one unique solution, then please describe the solutions.)
4x + 10y = 6
5
2
x- y=1
3
3
Page 139
180
25) Solve the following system of equations algebraically. (If the system does not have a solution, then say so or if it
has more than one unique solution, then please describe the solutions.)
x+y+z=2
x + 2y + 3z = 4
x + 3y + 5z = 7
181
26) Solve the following system of equations algebraically. (If the system does not have a solution, then say so or if it
has more than one unique solution, then please describe the solutions.)
x+y+z=2
x + 2y + 3z = 4
x + 3y + 5z = 6
182
27) Solve the following questions algebraically:
183
184
185
186
187
2p + 3q = 5
2p + 6q = 10
28) A business woman has $300,000 of profits from her office supply company invested in two investments. One
has a yearly return of 6% and the other has a yearly return of 7%. If the total yearly income from the
investments is $19,300, how much is invested at each rate?
29) A retired couple has $500,000 invested in two bond funds that earn 5% and 7%. If the total yearly income from
the investments is $30,000, how much is invested at each rate?
30) A young family with two children has $40,000 saved for college costs, with part invested at 12% and part
invested at 8%. If the total yearly income from the investments is $3400, how much is invested at each rate?
31) 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.
32) 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?
188
33) Two types of chickens, A and B, are raised in a chicken coop. Each day they receive 1.4 kilograms of corn and
2.4 kilograms of millet. Each chicken of type A requires 100 grams of corn and 150 grams of millet. Each
chicken of type B requires 60 grams of corn and 120 grams of millet. How many of each type of chicken will the
corn and millet support so that all of the food is consumed each day? Use a graphing calculator to solve the
system by finding the point of intersection of the two equations.
189
34) 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?
Page 140
190
35) 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 700 grams of the second supplement. Each
monkey of species A requires 25 g of the first supplement and 50 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?
191
36) Two species of deer, A and B, living in a wildlife refuge are given extra food in the winter. Each week they
receive 3.5 tons of food pellets and 7 tons of hay. Each deer of species A requires 4 pounds of the pellets and 8
pounds of hay. Each deer of species B requires 2 pounds of the pellets and 4 pounds of hay. How many of each
species of deer will the food support so that all of the food is consumed each week? Use a graphing calculator
to graph both of your equations at the same time. What do you notice about the graphs?
192
37) Two types of chickens, A and B, are raised in a chicken coop. Each day they receive 1.4 kilograms of corn and
2.8 kilograms of millet. Each chicken of type A requires 100 grams of corn and 200 grams of millet. Each
chicken of type B requires 60 grams of corn and 120 grams of millet. How many of each type of chicken will the
corn and millet support so that all of the food is consumed each day?
193
38) 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?
194
195
39) A chemist has two solutions that contain different concentrations of hydrochloric acid. One is a 20%
concentration and the other is a 12% concentration. How many cubic centimeters of each should he mix to
obtain 100 cc with a concentration of 15.2%?
40) A swimming pool owner has two solutions which contain different concentrations of chlorine. One is a 5%
solution and the other is a 10% solution. How many liters of each should she mix to obtain 8 liters with a 6%
concentration?
196
41) A nut shop packages mixtures of different nuts for sale. From peanuts, cashews and almonds, the owner wants
to prepare a mixture which will sell for $4.45 for a 1 pound bag. The cost per pound of these nuts is $1.50, $6.00,
and $4.00, respectively. The amount of peanuts is to be three times the amount of almonds. How much of each
type of nut will be in the final blend?
197
42) A hobby store packages mixtures of different beads for sale. From wood beads, glass beads, and metal beads,
the owner wants to prepare a mixture which will sell for $4.20 for a 200 count bag. The cost per bead of these
beads is $0.01, $0.03, and $0.05, respectively. The number of wooden beads is to be four times the number of
metal beads. How many of each type of bead will be in the final package?
3.5 Nonlinear Systems
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
198
2
1) Solve the system: y = 4x - x
y = x2 - 6
199
2) Solve the system:
x2 + y - 3 = 0
2x + y = 0
Page 141
200
3) Solve the system:
y= x-4
x-y=4
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
201
4) One solution of the system
A) x =
9
3
,y=25
5
B) x =
25
5
,y=9
3
C) x =
49
7
,y=2
2
x - y2 = 0
3x + 2y - 5 = 0
is x = 1 and y = 1. Another solution is
D) x = 1, y = -1
E) x = 0, y = 0
202
5) An x-value of a solution of the system
A) -1.
203
B) 1.
6) The number of solutions of the system
A) zero.
B) one.
x-y-1=0
is
y= x+5
C) 4.
D) -4.
E) 11.
D) three.
E) four.
x2 + y2 = 7
is
x2 - y2 = 1
C) two.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
204
205
7) Solve the following nonlinear system:
y = 8 - x2
4x - y + 11 = 0
8) Solve the following nonlinear system of equations by sketching its graph (using graphing calculator or using
graph paper.) If the equations have no solutions, please say so.
y = x 2 - 2x + 5
y = -x
206
9) The shape of the main cable on a suspension bridge can be described by the function y =
1 2
1
x +
x + 10,
500
250
where y is the height of the cable (in feet) above the roadbed and x is the horizontal distance (in feet) from the
center of the bridge. A cross support passes by the cable, and its position can be described by the function
y = 0.2x + 9.8. Where does the cross-support intersect the cable? Use a graphing calculator to answer the
question by finding the point(s) of intersection of the two equations.
207
10) The shape of a rope bridge stretched across a ravine can be described by the function y = 0.003x 2 + 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.
Page 142
208
11) Solve the system
y = 2x 2 - 5
x2 + y = 4
209
12) Solve the system
2x - 5y = 10
y= x+4
3.6 Applications of Systems of Equations
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
210
1) Suppose the supply and demand equations for a manufacturerʹs product are p =
1
3
q + 6 and p = q + 14,
50
100
respectively, where q represents number of units and p represents price per unit in dollars. Determine (a) the
equilibrium quantity; (b) the equilibrium price. If a tax of $1.00 per unit is imposed on the manufacturer, (c)
determine the new equilibrium quantity; (d) the new equilibrium price.
211
2) 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?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
212
3) 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
A) 3000 units.
213
B) 3500 units.
C) 4000 units.
4) The supply and demand equations for a product are p =
D) 4500 units.
E) 5000 units.
1
1
q + 20 and p = 200 - q, respectively, where q
2
10
represents the number of units and p represents the price per unit in dollars. The equilibrium price is
A) $10.
B) $20.
C) $30.
D) $40.
E) $50.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
q
- 3 and the supply equation is
20
214
5) Find the equilibrium point if the demand equation for a product is p =
215
80
p=
1
1
6) Suppose that the supply and demand equations for a certain product are p =
q - 9 and p = q + 3,
70
14
respectively, where p represents the price per unit in dollars and q represents the number of units per time
period.
(a) Find the equilibrium price algebraically.
(b) Find the equilibrium price when a tax of 50 cents per unit is imposed.
216
7) Find the Break Even Quantity for a product whose Total Revenue, y TR, (in $) and Total Cost, y TC , (in $) are as
follows:
y TR = (10q - 25)q
y TC = 2000 + 75q
Page 143
217
8) Find the Break Even Point for a product whose Total Revenue, y TR, (in $) and Total Cost, y TC , (in $) are as
follows:
y TR = (10q - 25)q
y TC = 2000 + 75q
218
219
9) Find the Equilibrium Quantity for a product with Demand Equation and Supply Equation as follows:
Demand: p = 300 - 8q
19
Supply:
p=
q+5
5
10) Find the Point of Equilibrium for a product with Demand Equation and Supply Equation as follows:
Demand: p = 300 - 8q
19
q+5
Supply:
p=
5
Page 144