Exponential Functions
June 18, 2012
Linear and Exponential Equations
Linear:
 For a ONE unit increase in the input, a constant number is _______________ to the output.

The general form of a linear equation is:
Exponential:
 For a ONE unit increase in the input, a constant number is _______________ to the output.

The general form of an exponential equation is:
1. Use the following functions to answer the questions below:
A  t   1000  50t
B  t   200 1.5 
D  t   4t 2  300
E  t   5000
t
C  t   300  0.5 
F t  
t
800
t
a. Which of the above functions are linear?
b. For those functions listed in part a, identify the slope and vertical intercept.
c. Which of the above functions are exponential?
d. For those functions listed in part c, identify the factor and vertical intercept.
2. Using your calculator, evaluate the exponential functions given below when t  1 .
a.
F  t   100 1.15 
F 1 
t
5
G  t   100  
2
G 1 
t
H  t   100  4.3 
t
H 1 
b. Conclusion: When the value of the factor is greater than 1  a  1 , the exponential function is
(growing / decaying).
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Exponential Functions
June 18, 2012
c.
F  t   100  0.15 
t
 1
G  t   100  
4
F 1 
H  t   100  0.60 
t
G 1 
t
H 1 
d. Conclusion: When the value of the factor is between 0 and 1  0  a  1 , the exponential
function is (growing / decaying).
3. Determine if the function from the table is linear, exponential, or neither.
t
F(t)
G(t)
H(t)
J(t)
K(t)
L(t)
M(t)
0
10
10
10
10
10
10
10
1
30
15
12.5
11
15
5
10
2
90
20
15
14
22.5
2.5
10
3
270
25
17.5
19
33.75
1.25
10

F(t) is _______________ because for a one unit increase in t, the function is
_______________ by a constant value of _____.

G(t) is _______________ because for a one unit increase in t, the function is
_______________ by a constant value of _____.

H(t) is _______________ because for a one unit increase in t, the function is
_______________ by a constant value of _____.

J(t) is _______________ because for a one unit increase in t, the function is
_______________ by a constant value of _____.

K(t) is _______________ becase for a one unit increase in t, the function is
_______________ by a constant value of _____.

L(t) is _______________ because for a one unit increase in t, the function is
_______________ by a constant value of _____.

M(t) is _______________ because for a one unit increase in t, the function is
_______________ by a constant value of _____.
4. Write the equation for all linear and exponential function(s) from #3.
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Exponential Functions
June 18, 2012
5. When Vail Mountain opened in 1962, a lift ticket cost $5. Since then, prices have approximately
doubled every 10 years.
a. Create a table for this situation.
Year
Cost in
Dollars
1962
1972
1982
1992
2002
b. Would linear or exponential growth better model this relationship? Explain.
c. Define the variables for this situation.
d. Create a graph that models this situation.
90
80
70
60
50
40
30
20
10
10
20
30
40
e. Create an equation to model the cost of a lift ticket as a function of the time in years after 1962.
Round to 3 decimal places.
3
Exponential Functions
June 18, 2012
6. Depending on the aggressiveness of a breast tumor, its volume can double in weeks or months.
On average, the volume of a breast tumor doubles every 100 days. Using an initial volume of 0.06
cm3 (the minimal tumor volume detectable by a mammogram) and measuring time in days after the
tumor reached that initial volume,
a. Identify the independent and dependent variables.
b.
Create an exponential function describing this situation. Round to 3 decimals.
7. The concentration in the blood stream of theophylline, a common asthma drug, can be modeled by
an exponential model. Let T be the concentration of theophylline in mg/L and let h be the number
of hours.
a. Using the information that the initial concentration is 12 mg/L and 6 hours later, the
concentration is 4.2 mg/L, write an exponential function of T as a function of h. Round to 3
decimals.
b. Evaluate T(9) AND interpret this value in the context of the situation.
4