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EXAMPLE 1 Write an exponential function
Write an exponential function y = ab whose graph passes through (1, 12) and (3, 108).
x
SOLUTION
STEP 1 Substitute the coordinates of the two given points into y = ab .
x
12 = ab1
108 = ab3
Substitute 12 for y and 1 for x.
Substitute 108 for y and 3 for x.
STEP 2 Solve for a in the first equation to obtain
a = , and substitute this expression for
a in the second equation.
12 b
EXAMPLE 1 Write an exponential function
108 = b312 b
108 = 12b2
29 = b
3 = b
Substitute for a in second equation.
12 b
Simplify.
Divide each side by 12.
Take the positive square root because b > 0.
STEP 3 Determine that a = 12 b = 12
3 = 4. so, y = 4 3 .x
EXAMPLE 2 Find an exponential model
• Draw a scatter plot of the data pairs (x, ln y). Is an exponential model a good fit for the original data pairs (x, y)?
• Find an exponential model for the original data.
A store sells motor scooters. The table shows the number y of scooters sold during the xth year that the store has been open.
Scooters
EXAMPLE 2 Find an exponential model
SOLUTION
Use a calculator to create a table of data pairs (x, ln y).
STEP 1
Plot the new points as shown. The points lie close to a line, so an exponential model should be a good fit for the original data.
STEP 2
x 1 2 3 4 5 6 7
ln y 2.48 2.77 3.22 3.58 3.91 4.29 4.56
EXAMPLE 2 Find an exponential model
STEP 3
Find an exponential model y = ab by choosing two points on the line, such as (1, 2.48) and (7, 4.56). Use these points to write an equation of the line. Then solve for y.
x
ln y – 2.48 = 0.35(x – 1)
ln y = 0.35x + 2.13
y = e0.35x + 2.13
y = e (e )2.13 0.35 x
y = 8.41(1.42) x
Equation of line
Simplify.
Exponentiate each side using base e.
Use properties of exponents.
Exponential model
EXAMPLE 3 Use exponential regression
SOLUTION
Use a graphing calculator to find an exponential model for the data in Example 2. Predict the number of scooters sold in the eighth year.
Scooters
Enter the original data into a graphing calculator and perform an exponential regression. The model is y = 8.46(1.42) .x
Substituting x = 8 (for year 8) into the model gives y = 8.46(1.42) 140 scooters sold.
8
GUIDED PRACTICE for Examples 1, 2 and 3
Write an exponential function y = ab whose graph passes through the given points.
x
1. (1, 6), (3, 24)
SOLUTION
STEP 1 Substitute the coordinates of the two given points into y = ab .
x
6 = ab1
24 = ab3
Substitute 6 for y and 1 for x.
Substitute 24 for y and 3 for x.
GUIDED PRACTICE for Examples 1, 2 and 3
STEP 2 Solve for a in the first equation to obtain
a = , and substitute this expression for
a in the second equation.
6 b
24 =b3
6 b
24 = 6b2
24 = b
2 = b
Substitute for a in second equation.
6 b
Simplify.
Divide each side by 6.
Take the positive square root because b > 0.
STEP 3 Determine that a = 6 b = 6
2 = 3. so, y = 3 2 .x
GUIDED PRACTICE for Examples 1, 2 and 3
Write an exponential function y = ab whose graph passes through the given points.
x
2. (2, 8), (3, 32)
STEP 1 Substitute the coordinates of the two given points into y = ab .
x
8 = ab1
32 = ab3
Substitute 8 for y and 2 for x.
Substitute 32 for y and 3 for x.
SOLUTION
GUIDED PRACTICE for Examples 1, 2 and 3
STEP 2
32 = 8b
4 = b
Divide each side by 4.
Take the positive square root because b > 0.
STEP 3
Solve for a in the first equation to obtain
a = , and substitute this expression for
a in the second equation.
8 b2
32 =b3
8 b2
Substitute for a in second equation.
8 b2
Determine that a = = 8 b2
8 42
816
1 2== .
1 2So, y = 4x
GUIDED PRACTICE for Examples 1, 2 and 3
Write an exponential function y = ab whose graph passes through the given points.
x
3. (3, 8), (6, 64)
STEP 1 Substitute the coordinates of the two given points into y = ab .
x
8 = ab3
64 = ab6
Substitute 8 for y and 3 for x.
Substitute 64 for y and 6 for x.
SOLUTION
GUIDED PRACTICE for Examples 1, 2 and 3
STEP 2
b = 2
Divide each side by 8.
Take the positive square root because b > 0.
Solve for a in the first equation to obtain
a = , and substitute this expression for
a in the second equation.
8 b3
64 =b6
8 b3
Substitute for a in second equation.
8 b3
64= 8b3
b = = 83 64 8 Simplify.
GUIDED PRACTICE for Examples 1, 2 and 3
STEP 3 8 b3Determine that a = = 8
23 8 8 == .1
So, b = 2, a = 1, y = 1 2 = 2 .x x
GUIDED PRACTICE for Examples 1, 2 and 3
SOLUTION
4. WHAT IF? In Examples 2 and 3, how would the exponential models change if the scooter sales were as shown in the table below?
The initial amount would change to 11.39 and the growth rate to 1.45.
