Name _____________________________
Linear and Exponential Models
Study Island
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1. A company is holding a dinner reception in a
hotel ballroom. The graph represents the total cost
of the ballroom rental and dinner.
3. In an organism whose environment does not
affect cell division, a parent cell divides into two
daughter cells. Those two daughter cells each divide
into two more cells for a total of four cells in the
organism. If those four cells each divide into two
more cells, there will be eight cells in the organism.
If each of these cells continue to divide in the same
manner, how many cells will be in the organism
after fourteen divisions?
A. 268,435,456
B. 8,192
C. 16,384
D. 28
4. Which function family would be used to solve the
following question?
Which function represents the data displayed in the
graph?
f(x) = $7.50x + $625.00
A.
B.
C.
D.
f(x) = $625.00x + $7.50
f(x) = $0.75x + $625.00
A rabbit population doubles every year. If the rabbit
population starts with 5 rabbits, then how many will
there be in 6 years?
A. Absolute Value Function Family
B. Quadratic Function Family
C. Exponential Function Family
D. Linear Function Family
f(x) = $625.00x + $0.75
2. The monthly cost of operation at a company, C,
given in dollars as a function of the number of units
produced per month, u, is given below.
5. A pan is heated to 433°F, then removed from the
heat and allowed to cool in a kitchen where the
room temperature is a constant 71°F. The formula
below can be used to find D, the difference in
temperature between the pan and the room after t
minutes.
C = $3,173 + $31u
D = 362e-0.03t
If the company wants to keep the cost of operation
under $6,000 per month, what is the maximum
number of units they can produce?
A. 91
B. 92
C. 910
D. 911
What is the approximate temperature of the pan
after it has been away from the heat for 20 minutes?
A. 127.7°F
B. 198.7°F
C. 730.6°F
D. 269.7°F
Name _____________________________
6. A contract delivery worker is paid $12.00 per
hour as well as $0.50 per mile. Assuming a 40-hour
work week, which of the following equations below
gives the amount of money per week that the
worker makes before taxes, M, as a function of the
number of miles, m, driven during the week?
Linear and Exponential Models
9. The graph below represents the bacteria
population after t minutes.
A. M = 480 + 20m
B. M = 12.5m
C. M = 12 + 20m
D. M = 480 + 0.5m
7. A car was originally purchased for $25,000. The
following equation gives the car's value when it has
been driven a total of m thousand miles over a
period of t years.
If the car has been driven a total of 25,000 miles
over a period of 7 years, what is its value?
A. $3,661.30
B. $1,363.79
C. $7,377.26
D. $3,037.82
8. The table below represents a linear situation.
x
f(x)
4
-19
5
-23
6
-27
7
-31
Construct the function which represents the data
displayed in the graph.
A. P(t) = 200(2)t
B. P(t) = 50(4)t
C. P(t) = 50(2)t
D. P(t) = 4t + 50
10. In order to pay back a debt, Mark has set up a
bank account. He has asked that 11% of the total
amount in the account be withdrawn each month as
a payment towards his debt. If Mark started the
account with $8,961.00, and has not made any
further deposits, what will the approximate balance
in the bank account be after 14 months?
(Hint: Use the formula for depreciation, y = A(1 r)t, where A is the initial amount in the account, r is
the withdrawal rate in percentage terms, and t is the
amount of time that has passed in months.)
A. $1,772.70
Use this table to construct the function that it
represents.
B. $7,975.29
C. $1,752.77
D. $1,560.29
A. f(x) = -3x - 4
B. f(x) = -4x - 3
C. f(x) = -4x - 23
D. f(x) = 4x + 15
Answers
Name _____________________________
1. A
2. A
3. C
4. C
5. D
6. D
7. D
8. B
9. B
10. C
Explanations
1. The graph is a linear function represented by an
equation of the form f(x) = mx + b, where m is the
rate of change and b is the initial value.
First, calculate the rate of change, or slope, between
any two points on the graph. In this case, use the
points (10, 700) and (30, 850).
Linear and Exponential Models
under $6,000 and a fraction of a unit cannot be
made, the maximum number of units they can
produce is 91.
3. After zero cell divisions, there is one cell in the
organism.
1 = 20
After one cell division, there are two cells in the
organism.
2 = 21
After two cell divisions, there are four cells in the
organism.
4 = 22
After three cell divisions, there are eight cells in the
organism.
8 = 23
After x cell divisions, the number of cells in the
organism can be represented by the following
equation.
f(x) = 2x
Therefore, after fourteen cell divisions, there will be
214 = 16,384 cells in the organism.
Next, determine the initial value. The initial value is
the value of f(x) when x equals 0.
According to the graph, the initial value is $625.00,
which is the rental fee for the ballroom.
Therefore, the function that represents the graph is
f(x) = $7.50x + $625.00.
2. To determine the maximum number of units,
substitute $6,000 in for C in the given equation, and
then solve for u.
$6,000 = $3,173 + $31u
$2,827 = $31u
91.194 ≈ u
4. Growth that doubles every year is always
modeled by an exponential function.
Since the problem states that the rabbit population
doubles every year, then an equation in the
exponential function family would be used to
solve this question.
5. First, substitute t = 20 into the formula and solve
for D.
D = 362e-0.03(20)
D 198.7°F
Next, since D is the difference between the
temperatures of the pan and of the room, add D to
Since the company wants their operating cost to be
Name _____________________________
the temperature of the room to find the temperature
of the pan.
71°F + 198.7°F = 269.7°F
6. First, calculate the amount of money the worker
will make given a 40-hour work week at $12.00 per
hour.
(40)($12.00) = $480.00
Then, add this to the amount of money that the
worker will get paid per mile to get the total weekly
earnings.
$480.00 + $0.50m
Therefore, the equation that gives the amount of
money per week that the worker makes before
taxes, M, assuming a 40-hour work week as a
function of the number of miles, m, driven during
the week is given below.
Linear and Exponential Models
-31 - (-27) = -4
So, the slope is -4.
The y-intercept occurs at the point where x is zero.
The table does not show an x-value of zero. Instead,
use the point-slope form of a line, shown below,
where (x1, y1) represents a point on the line, and m
represents the slope.
(y - y1) = m(x - x1)
Substitute the point (4, -19) and the slope, -4, into
the equation. Then, transform the equation so that it
is in slope-intercept form.
(y - (-19)) = -4(x - 4)
y + 19 = -4x + 16
y = -4x + 16 - 19
y = -4x - 3
M = 480 + 0.5m
Therefore, the function f(x) = -4x - 3 represents the
linear situation in the table.
7. First, decide what the variables m and t are equal
to.
9. Exponential functions are of the form f(x) = a(b)x,
where b is greater than zero and not equal to one.
Since the car has been driven 25,000 miles, m = 25.
Since the car has been driven for 7 years, t = 7.
Next, substitute these values into the given
equation, and then use the order of operations to
solve for V.
Calculate the common ratio. The calculations below
use the points (0, 50), (0.5, 100), (1, 200), (1.5,
400), (2, 800), and (2.5, 1,600).
The change in the t-values is 0.5, so divide 2 by 0.5.
8. A linear function can be represented by an
equation in slope-intercept form, shown below,
where m represents the slope, and b represents the
y-intercept.
f(x) = mx + b
First, determine the slope of the line. Notice that the
x-values increase by 1. The difference between the
f(x)-values will reveal the slope.
-23 - (-19) = -4
-27 - (-23) = -4
So, the common ratio is 4.
Thus, the exponential function will have the form
P(t) = a(4)t.
Use the point (0, 50) to find the value of a.
Therefore, the function that represents the data in
the graph is P(t) = 50(4)t.
Name _____________________________
10. Evaluate the depreciation formula with the
given values of A, r, and t.
Therefore, after 14 months, the approximate balance
of the bank account will be $1,752.77.
Linear and Exponential Models