4.7 Write and Apply Exponential & Power Functions

1. 4.7 Write and Apply Exponential & Power Functions p. 509 How do you write an exponential function given two points? How do you write a power function given two points? Which function uses logs to solve it?

2. Just like 2 points determine a line, 2 points determine an exponential curve.An Exponential Function is in the form of y=abx

3. Write an Exponential function, y=abx whose graph goes thru (1,6) & (3,24) • Substitute the coordinates into y=abx to get 2 equations. • 1. 6=ab1 • 2. 24=ab3 • Then solve the system:

4. Write an Exponential function, y=abx whose graph goes thru (1,6) & (3,24) (continued) • 1. 6=ab1→ a=6/b • 2. 24=(6/b) b3 • 24=6b2 • 4=b2 • 2=b a= 6/b = 6/2 = 3 So the function is Y=3·2x

5. 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 SOLUTION STEP 1 Substitute 12 for yand 1 for x. Substitute 108 for yand 3 for x. STEP 2

6. Substitute for a in second equation. 108 = b3 12 12 12 b 3 b x Determine that a = = = 4 so, y = 4 3 . 12 b Simplify. Divide each side by 12. Take the positive square root because b > 0. STEP 3

7. Write an Exponential function, y=abx whose graph goes thru (-1,.0625) & (2,32) • .0625=ab-1 • 32=ab2 • (.0625)=a/b • b(.0625)=a • 32=[b(.0625)]b2 • 32=.0625b3 • 512=b3 • b=8 y=1/2 · 8x a=1/2

8. Power Function A Power Function is in the form of y = axb Because there are only two constants (a and b), only two points are needed to determine a power curve through the points

9. Modeling with POWER functions a = 5/2b 9 = (5/2b)6b 9 = 5·3b 1.8 = 3b log31.8 = log33b .535 ≈ b a = 3.45 y = 3.45x.535 • y = axb • Only 2 points are needed • (2,5) & (6,9) • 5 = a 2b • 9 = a 6b

10. 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 SOLUTION STEP 1 Substitute 2 for yand 3 forx. Substitute 9 for yand 6 forx.

11. 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 b Substitute for ain second equation. 9 = 6 b 3 Log2 2 3 b Log 4.5 = b 2 2 b Take log of each side. 9 = 2 2 b 4.5 = 2 2 b 3 = b 2.17 b STEP 2 Simplify. Divide each side by 2. Change-of-base formula Use a calculator. STEP 3

12. b Write a power function y = ax whose graph passes through the given points. Substitute the coordinates of the two given points into y = ax. b b 4 = a 3 b 15 = a 6 6. (3, 4), (6, 15) SOLUTION STEP 1 Substitute 4 for yand 3 forx. Substitute 15 for yand 6 forx.

13. Solve for ain the first equation to obtain a= , and substitute this expression for ain the second equation. b 15 = 6 4 15 Substitute for ain second equation. b 3 4 b 15 = 4 2 b = 2 4 3 b Log 3.7 = b 2 Take log of each side. 4 2 3 b STEP 2 Simplify. Divide each side by 4. 3.7 = 2

14. 4 3 1.9 1.91 = 1.9 Determine that a = 0.492. So, y = 0.492x. Log 3.7 0.5682 Log2 0.3010 = b 1.90 b Change-of-base formula Simplify. Use a calculator. STEP 3

15. 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. Biology page 284 • 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. How do you write an exponential function given two points? y = abx • How do you write a power function given two points? y = axb • Which function uses logs to solve it? Power function y = axb

17. Homework • Page 285 • 3-7, 15-19