90)
Sketch the graph of 5(x+ 3) – 3(y– 1) = 0.
90)
91)
Find the slope of the line passing through the points (5, –3) and (2, –1).
91)
92)
Show that the points A(–1, 2), B(3, –2), and C(–6, –3), are the vertices of a right triangle.
92)
93)
Find an equation of the horizontal line that passes through the point (5, 6).
93)
94)
Find the range of the function y=f(x) = – x2+ 3x+ 2.
94)
21
95)
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?
95)
96)
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?
96)
97)
State whether f(x) = 10 + 16x– 4x2 has maximum or minimum value and find that value.
97)
98)
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?
98)
99)
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
99)
100)
What is the slope of a horizontal line?
100)
101)
State whether f(x) = 12x2– 24x+ 10 has maximum or minimum value and find that value.
101)
102)
Find the equation of a line which is perpendicular to the line x= 0.521 and passes through
the point (0.1, –7).
102)
103)
Find the x–intercept made by the line that passes through the points (1, –1) and (–1, 0).
103)
104)
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?
104)
105)
For the linear function f(x) = 2x+ 1, find: (a) the slope and (b) the vertical axis intercept. (c)
Sketch the graph of f.
105)
106)
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.
106)
107)
Solve the following system algebraically:
2x–y+ 3z=12
x+y–z= – 3
x + 2y– 3z= – 10
107)
108)
Determine whether the following lines are parallel, perpendicular or neither.
3x– 4y= 3
12x+ 6y= 12
108)
109)
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.
109)
110)
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?
110)
111)
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.
111)
112)
The daily profit for a lighting store from the sale of a table lamp is given by
P(x) =–x2+ 34x+ 35, where x is the number of table lamps sold. Use a graphing calculator
to graph the function.
112)
113)
State whether the following function has a maximum value or minimum value and find it:
f(x) = 2 + 10x–x2
113)
114)
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?
114)
115)
Solve the system: y=x– 4
x–y= 4
115)
116)
Find the equation of a line which is perpendicular to the line 2x+ 3y– 57 = 0 and passes
through the point (1, –1).
116)
117)
Solve the following system algebraically: 5x+ 2y= 36
8x– 3y= – 54
117)
118)
The demand function for a manufacturer’s 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 manufacturer’s total revenue and determine this revenue.
118)
119)
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.
119)
120)
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
120)
121)
Solve the system 2x– 5y= 10
y=x + 4
121)
122)
The shape of the main cable on a suspension bridge can be described by the function
y=1
500x2+1
250x+ 10, 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.
122)
123)
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.
123)
124)
Find an equation of the line that passes through the origin and that has slope –5.
124)
125)
Solve the following system algebraically: 8x– 4y=7
y =2x– 4
125)
126)
Find the Break Even Quantity for a product whose Total Revenue, yTR, (in $) and Total
Cost, yTC, (in $) are as follows:
yTR = (10q– 25)q
yTC = 2000 + 75q
126)
127)
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.
127)
128)
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?
128)
129)
Solve the following system algebraically: 5u+v= – 2
20u+2v= 1
129)
27
130)
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.
130)
131)
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.
131)
132)
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?
132)
133)
Suppose that f(x) is a linear function with slope = – 3 and y–intercept 1
5, then find the
function f(x).
133)
134)
Solve the following system algebraically: 3x+ 5y= – 6
2x– 6 =5y
134)
135)
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
2
3x–5
3y= 1
135)
136)
Suppose f is a linear function with slope 5 and such that f(1) = 4. Find f(x).
136)
137)
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.
137)
138)
Graph the function y=f(x) = 3 – 2x–x2 and indicate the coordinates of the vertex and
intercepts.
138)
139)
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.
139)
140)
Suppose that the vertex of the parabola y= 3x2– 6x+k is (1, 2); find k.
140)
141)
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?
141)
30
142)
Graph the equation 5x +y+ 8 = 0.
142)
143)
The relationship between temperature on the Fahrenheit scale and the temperature on the
Celsius scale is C=5
9(F– 32). Find the slope and y–intercept of the equation.
143)
144)
Determine an equation of the vertical line that passes through the point (3, –6).
144)
145)
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?
145)
146)
The daily profit for the garden department of a store from the sale of trees is given by P(x)
=–x2+ 18x+ 144, where x is the number of trees sold. Use a graphing calculator to graph
the function.
146)
147)
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?
147)
148)
Graph the general linear form of the Olympic discus throw equation whose slope–intercept
form is d= 1.75t+ 175.
148)
149)
Find the equation of a line having slope 0.5 and x–intercept = 1.3.
149)
150)
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.
150)
151)
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.
151)
152)
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.
152)
153)
Graph the general linear form of the Fahrenheit–Celsius conversion equation whose
slope–intercept form is F=9
5C+ 32.
153)
154)
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?
154)
155)
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
2
3x–5
3y= 7
155)
156)
Graph the function y=f(x) =x2– 6x+ 5 and indicate the coordinates of the vertex and
intercepts.
156)
157)
Find a general linear equation of the line that passes through point (–6, 4) and has slope –2.
157)
158)
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%?
158)
159)
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=x2– 2x+ 5
y= – x
159)
160)
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 noise levels, k= 4. Use a graphing calculator to graph the
equation.
160)
161)
Solve the system: x2+ y – 3 = 0
2x + y = 0
161)
162)
Determine whether the following lines are parallel, perpendicular or neither.
3x– 2y= – 1
2x+ 3y= – 20
162)
163)
Suppose f(p) is a linear function given by f(p) =3p– 2
5. Find the slope and the y–intercept.
163)
164)
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.)
164)
165)
Solve the following system algebraically:
2x+y+z = 0
4x+ 3y+ 2z= 2
2x –y– 3z= 0
165)
166)
The demand function for an appliance company’s 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 manufacturer’s total
revenue, and determine this revenue.
166)
167)
Find the vertex and axis of symmetry of the parabola y= – x2+ 6x+ 3. Also find if it opens
upward or downward.
167)
168)
Determine the equation of the line which is perpendicular to 2x–y + 3 = 0 and has
y–intercept 6.
168)
169)
Find the equation of the line with y–intercept 4 and slope –2
3.
169)
170)
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 odor intensity, k= 5. Use a graphing calculator to graph
the equation.
170)
171)
Solve the following system algebraically: 2x–y= 1
–x+ 2y= 7
171)
172)
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?
172)
173)
Find the equation of a line which is parallel to the line x= 0.521 and passes through the
point (0.1, –7).
173)
174)
Determine the linear function f(t) with slope = – 1 and f(2) = 1.
174)
175)
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.
175)
176)
The equation of a certain line is y+ 2 = – 2(x– 3). Find: (a) the slope–intercept form and (b)
a general linear form.
176)
177)
Graph the function y=f(x) =x2– 6x and indicate the coordinates of the vertex and
intercepts.
177)
178)
Determine whether the following lines are parallel, perpendicular or neither.
12x+ 4y= 16
24x+ 8y= 36
178)
179)
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.
179)
180)
Find the Point of Equilibrium for a product with Demand Equation and Supply Equation
as follows:
Demand: p= 300 – 8q
Supply: p=19
5q + 5
180)
181)
Find the equation of a line having slope = 1 and x–intercept = – 2.
181)
182)
Suppose f is a linear function such that f(–2) = 5 and f(5) = 2. Find f(x).
182)
183)
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?
183)
184)
For the linear function f(x) = – 5x+ 5, find: (a) the slope and (b) the vertical axis intercept.
(c) Sketch the graph of f.
184)
185)
Solve the following nonlinear system: y=8 –x2
4x–y+ 11 =0
185)