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Algebra 2:

Algebra 2 Chapter 7 Lesson 7

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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 = ab 1 108 = ab 3 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

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EXAMPLE 1 Write an exponential function 108 = b 3 12 b 108 = 12 b 2 2 9 = 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

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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 x th year that the store has been open. Scooters

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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

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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 = 0.35 x + 2.13 y = e 0.35 x + 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

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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

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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 y = 3 2 x 2. (2, 8), (3, 32) SOLUTION 1 2 y = 4 x 3. (3, 8), (6, 64) SOLUTION y = 2 x

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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.

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EXAMPLE 4 Write a power function Write a power function y = ax whose graph passes through (3, 2) and (6, 9) . b b 2 = a 3 9 = a 6 b Substitute 2 for y and 3 for x . Substitute 9 for y and 6 for x . STEP 1 SOLUTION Substitute the coordinates of the two given points into y = ax . b

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EXAMPLE 4 Write a power function STEP 2 Solve for a in the first equation to obtain a = , and substitute this expression for a in the second equation. 2 3 b 9 = 6 2 3 b b 9 = 2 2 b 4.5 = 2 b Log 4.5 = b 2 2.17 b Substitute for a in second equation. 2 3 b Simplify. Divide each side by 2. Take log of each side. 2 Change-of-base formula Use a calculator. Log 4.5 Log2 = b

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EXAMPLE 4 Write a power function STEP 3 Determine that a = 0.184. So , y = 0.184 x . 2.17 3 2 2.17

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GUIDED PRACTICE for Example 4 Write a power function y = ax whose graph passes through the given points . b 5. (2, 1), (7, 6) y = 0.371 x 1.43 SOLUTION 6. (3, 4), (6, 15) y = 0.492 x 1.91 SOLUTION 7. (5, 8), (10, 34) y = 0.278 x 2.09 SOLUTION

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GUIDED PRACTICE for Example 4 8. REASONING Try using the method of Example 4 to find a power function whose graph passes through (3, 5) and (3, 7) . What can you conclude? SOLUTION The points cannot form a power function.

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EXAMPLE 5 Find a power model • Find a power model for the original data. The table at the right shows the typical wingspans x (in feet) and the typical weights y (in pounds) for several types of birds. Biology • Draw a scatter plot of the data pairs (ln x , ln y ). Is a power model a good fit for the original data pairs ( x , y ) ?

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EXAMPLE 5 Find a power model SOLUTION Use a calculator to create a table of data pairs (ln x , ln y ) . Plot the new points as shown. The points lie close to a line, so a power model should be a good fit for the original data. STEP 1 STEP 2

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EXAMPLE 5 Find a power model STEP 3 Find a power model y = ax b by choosing two points on the line, such as (1.227, 0.525) and (2.128, 2.774) . Use these points to write an equation of the line. Then solve for y . In y – y = m (In x – x ) 1 1 In y = 2.5 In x – 2.546 Equation when axes are ln x and ln y Substitute. Simplify. In y = In x – 2.546 2.5 Power property of logarithms In y – 2.774 = 2.5(In x – 2.128)

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EXAMPLE 5 Writing Reciprocals Y = e ln x – 2.546 2.5 Y = e e – 2.546 2.5 In x Y = 0.0784 x 2.5 Exponentiate each side using base e . Product of powers property Simplify.

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EXAMPLE 6 Use power regression Use a graphing calculator to find a power model for the data in Example 5 . Estimate the weight of a bird with a wingspan of 4.5 feet. Biology SOLUTION Enter the original data into a graphing calculator and perform a power regression. The model is y = 0.0442 x . 2.87 Substituting x = 4.5 into the model gives y = 0.0442( 4.5 ) 3.31 pounds. 2.87

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GUIDED PRACTICE for Example 5 and 6 9. The table below shows the atomic number x and the melting point y (in degrees Celsius) for the alkali metals. Find a power model for the data. SOLUTION y = 397.61 x -0.639