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Warm Up

Write and Apply Exponential and Power Functions. Warm Up. Lesson Presentation. Lesson Quiz. 60.4. ANSWER. ANSWER. y = –2 x + 9. y – 4.53 = 0.23( x – 2) or y – 5.22 = 0.23( x – 5). ANSWER. Warm-Up.

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Warm Up

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  1. Write and Apply Exponential and Power Functions Warm Up Lesson Presentation Lesson Quiz

  2. 60.4 ANSWER ANSWER y = –2x + 9 y – 4.53 = 0.23(x – 2) ory – 5.22 = 0.23(x – 5) ANSWER Warm-Up 1.Write an equation in slope-intercept form for the line through (2, 5) and (6, –3). 2.Write an equation in point-slope form for the line through (2, 4.53) and (5, 5.22). 3.What is the value of yif the point (10, y) is on the line y = 5.8x + 2.4?

  3. x Write an exponential function y = abwhose graph passes through (1, 12) and (3, 108). Substitute the coordinates of the two given points into y = ab . x 1 12 = ab 3 108 = ab Example 1 SOLUTION STEP 1 Substitute 12 for yand 1 for x. Substitute 108 for yand 3 for x.

  4. Substitute for ain second equation. 108 = b3 12 12 12 12 b 3 b b x Determine that a = = = 4.so, y = 4 3 . 108 = 12b 2 12 Solve for a in the first equation to obtain a= , and substitute this expression for ain the second equation. b 2 9 = b Example 1 STEP 2 Simplify. Divide each side by 12. 3 = b Take the positive square root because b > 0. STEP 3

  5. Scooters A store sells motor scooters. The table shows the number yof scooters sold during the xth year that the store has been open. Example 2 • 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.

  6. Example 2 SOLUTION STEP 1 Use a calculator to create a table of data pairs (x, ln y). STEP 2 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.

  7. x Find an exponential model y = abby 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. 0.35x+ 2.13 y = e x y = e(e ) 2.13 0.35 x y = 8.41(1.42) Example 2 STEP 3 ln y – 2.48 = 0.35(x – 1) Equation of line ln y = 0.35x + 2.13 Simplify. Exponentiate each side using base e. Use properties of exponents. Exponential model

  8. 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 Example 3 Scooters 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. SOLUTION

  9. x Write an exponential function y = abwhose graph passes through the given points. x y = 3 2 1 2 x 4 y = Guided Practice 1. (1, 6), (3, 24) SOLUTION 2. (2, 8), (3, 32) SOLUTION 3. (3, 8), (6, 64) x y = 2 SOLUTION

  10. 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? Guided Practice SOLUTION The initial amount would change to 11.39 and the growth rate to 1.45.

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

  12. Solve for ain the first equation to obtain a= , and substitute this expression for ain the second equation. 2 3 2.17 2.17 Determine that a = 0.184. So, y = 0.184x. 2 log 4.5 Substitute for ain second equation. b 3 log2 2 3 b log 4.5 = b 2 b Take log of each side. 2 9 = 2 2 b 4.5 = 2 2 3 b = b Example 4 STEP 2 b 9 = 6 Simplify. Divide each side by 2. Change-of-base formula STEP 3

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

  14. Guided Practice 8.REASONINGTry 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.

  15. Example 5 Biology The table at the right shows the typical wingspans x(in feet) and the typical weights y(in pounds) for several types of birds. • 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)? • Find a power model for the original data.

  16. Example 5 SOLUTION STEP 1 Use a calculator to create a table of data pairs (ln x, ln y). STEP 2 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.

  17. In y – y = m (In x – x ) 1 1 2.5 In y = In x – 2.546 Example 5 STEP 3 Find a power model y= axbby 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. Equation when axes are ln x and ln y In y – 2.774 = 2.5(In x – 2.128) Substitute. In y = 2.5 In x – 2.546 Simplify. Power property of logarithms

  18. 2.5 In x 2.5 Y = 0.0784x – 2.546 Y = e e Y = e 2.5 ln x– 2.546 Example 5 Exponentiate each side using base e. Product of powers property Simplify.

  19. Enter the original data into a graphing calculator and perform a power regression. The model is y =0.0442x . 2.87 Substituting x = 4.5into the model gives y = 0.0442(4.5) 3.31 pounds. 2.87 Example 6 Biology 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. SOLUTION

  20. -0.639 y =397.61x Guided Practice 9. The table below shows the atomic number xand the melting point y(in degrees Celsius) for the alkali metals. Find a power model for the data. SOLUTION

  21. Write an exponential functiony=abx Whose graphpasses through(2, 48) and(4, 768). 1. ANSWER y = 3•4x 2. Find an exponential model for the data in the table. ANSWER y = 19.4(1.81)x Lesson Quiz

  22. Write a power functiony=axbwhose graphpasses through(3, 8)and(6, 35). 3. ANSWER y = 0.77x2.13 Find a power model for the data in the table. ANSWER y = 0.08x2.3 Lesson Quiz 4.

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