EXAMPLE 1 Write an exponential function Write an exponential function y = ab whose graph passes...
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Transcript of EXAMPLE 1 Write an exponential function Write an exponential function y = ab whose graph passes...
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