#### Solved Examples and Worksheet for Fitting Exponential Models to Data

Q1Number of employees in an organization are shown in the table, where t represents time in years since 1991. Find the exponential model for the data.
 Years since 1991(t) 0 1 2 3 4 Number of employees 200 300 450 675 1012

A. R = 200(2.5)t
B. R = 300(0.75)t
C. R = 300(1.5)t
D. R = 200(1.5)t

Step: 1
Let the exponential function be R = M(N)t.
Step: 2
For t = 0, R = 200
Step: 3
200 = M(N)0
Step: 4
M = 200
Step: 5
For t = 1, R = 300 300 = 200N
Step: 6
N = 1.5
Step: 7
So, the exponential function that best fits the data is R = 200(1.5)t.
Correct Answer is :   R = 200(1.5)t
Q2Choose the exponential model for the data shown.
 x 0 1 2 3 4 f(x) 5 20 80 320 1280

A. f(x) = 5 · (45)x
B. f(x) = 5 · 4x
C. f(x) = 4 · 5x
D. f(x) = 5 · 4x

Q3Choose an exponential function for the table of values.
 x y -3 10 -2 100 -1 1000 0 10000 1 100000

A. y = 10000 · x10
B. y = 10000(x + 1)
C. y = 10000 · 10x
D. y = 10000 · 10- x

Step: 1
The table of values obtained by substituting x = -3, -2, -1, 0, 1 in the exponential function y = 10000 · 10x.
Correct Answer is :   y = 10000 · 10x
Q4Choose the exponential model for the data shown.
 x - 2 - 1 0 1 2 f(x) 89 43 2 3 92

A. f(x) = 2 · (32)x
B. f(x) = 2 · 3x
C. f(x) = 2 · 5x
D. f(x) = 2

Step: 1
Let the exponential function be f(x) = M (N)x.
Step: 2
At x = 0, f(x) = 2
[From the table.]
Step: 3
2 = M (N)0 M = 2
[Substitute the values in the exponential function given in step 1.]
Step: 4
At x = 1, f(x) = 3
[From the table.]
Step: 5
3 = M (N)1
[Substitute the values in the exponential function given in step 1.]
Step: 6
3 = 2N N = 32
[Substitute M = 2.]
Step: 7
Hence, the exponential function is f(x) = 2 · (32)x.
[Substitute the values of M & N in the exponential function given in step 1.]
Correct Answer is :   f(x) = 2 · (32)x
Q5George bought a house for $48000 in the year 2005. The following table shows the house value for the subsequent years. Choose the exponential model for the data shown.  Years since 2005(t) 0 1 2 3 4 value ($y) 55000 62150 70229.5 79359.3 89676

A. y = 55000(1.21)t
B. y = 55000(1.25)t
C. y = 55000(1.13)t
D. y = 55000(1.31)t

Step: 1
Let the exponential function be y = M(N)t.
Step: 2
At t = 0, y = 55000
[From the table.]
Step: 3
55000 = M(N)0
Step: 4
M = 55000
Step: 5
At, t = 1, y = 62150
Step: 6
62150 = 55000N
Step: 7
N = 1.13
[Simplify.]
Step: 8
So, the exponential function that best fits the data is y = 55000(1.13)t
[Substitute the values of M & N in the exponential function given in step 1.]
Correct Answer is :   y = 55000(1.13)t
Q6Profits of a company are shown in the table, where t represents time in years since 2006. Find the exponential model for the data.
 Yrs from 2006 (t) 0 1 2 3 4 Profit billion\$ (P) 9 12.15 16.4 22.14 29.89

A. P = - 9(1.35 )t
B. P = 16.40(1.49)t
C. P = - 12.15(1.49) t
D. P = 9(1.35)t

Step: 1
Let the exponential function be P = M(N)t.
Step: 2
For t = 0, P = 9
Step: 3
9 = M(N)0
Step: 4
M = 9
Step: 5
For t = 1, P = 12.15
Step: 6
12.15 = 9N
[from step 4.]
Step: 7
N = 1.35
[Simplify.]
Step: 8
So, the exponential function that best fits the data is P = 9(1.35)t
[Substitute the values of P, M and N in step 1.]
Correct Answer is :   P = 9(1.35)t
Q7Choose the exponential model for the data shown.

A. f(x) = 32(- 2)x
B. f(x) = 16( - 4)x
C. f(x) = 32(2)x
D. f(x) = 16(4)x

Step: 1
Let the exponential function be f(x) = M(N)x
Step: 2
At x = 0, f(x) = 16
[From the table.]
Step: 3
16 = M(N)0M = 16
[Substitute the values in the exponential function given in step 1.]
Step: 4
At x = 1, f(x) = 64.
[From the table.]
Step: 5
64 = M(N)1
[Substitute the values in the exponential function given in step 1.]
Step: 6
16N = 64 ⇒ N = 6416 = 4
[Substitute M = 16.]
Step: 7
Hence, the exponential function is f(x) = 16(4)x
[Substitute the values of M & N in the exponential function given in step 1.]
Correct Answer is :   f(x) = 16(4)x
Q8Choose the exponential model for the data shown.

A. f(x) = 18(8)x
B. f(x) = 14(8)x
C. f(x) = 18(2)x
D. f(x) = 18(4)x

Step: 1
Let the exponential function be f(x) = M(N)x
Step: 2
At x = 0, f(x) = 18.
[From the table.]
Step: 3
18 = M(N)0M = 18
[Substitute the values in the exponential function given in step 1.]
Step: 4
At x = 1, f(x) = 12
[From the table.]
Step: 5
12 = M(N)1
[Substitute the values in the exponential function given in step 1.]
Step: 6
18N = 12N = 82 = 4
[Substitute M = 18.]
Step: 7
Hence, the exponential function is f(x) = 18(4)x
[Substitute the values of M & N in the exponential function given in step 1.]
Correct Answer is :   f(x) = 18(4)x
Q9Revenues of a factory are shown in the table, where t represents time in years since 2006. Find the exponential model for the data.

A. R = 4(1.5)t
B. R = 4(0.75)2t
C. R = 4(1.25)t
D. R = 4(1.05)t

Step: 1
Let the exponential function be R = M(N)t
Step: 2
At t = 0, R = 4.
[From the table.]
Step: 3
4 = M(N)0M = 4
[Substitute the values in the exponential function given in step 1.]
Step: 4
At t = 1, R = 6.
[From the table.]
Step: 5
6 = M(N)1
[Substitute the values in the exponential function given in step 1.]
Step: 6
4N = 6 ⇒ N = 64 = 1.5
[Substitute M = 4.]
Step: 7
Hence, the exponential function is R = 4(1.5)t
[Substitute the values of M & N in the exponential function given in step 1.]
Correct Answer is :   R = 4(1.5)t
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