4.7 Write and Apply Exponential & Power Functions p. 509 How do you write an exponential function given two points? Just like 2 points determine a line, 2 points determine an exponential curve. An Exponential Function is in the form of y=abx 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: 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 a= 6/b = 6/2 = 3 • • • 24=6b2 4=b2 2=b So the function is Y=3·2x x Write an exponential function y =ab whose graph passes through (1, 12) and (3, 108). SOLUTION STEP 1 Substitute the coordinates of the two given points into y = ab x . Substitute 12 for y and 1 for x. 12 = ab 1 108 = ab 3 Substitute 108 for y and 3 for x. STEP 2 3 108 = 12 b b Substitute 12 b for a in second equation. Simplify. Divide each side by 12. Take the positive square root because b > 0. STEP 3 12 = 4 so, y = 4 3 x. Determine that a = 12 = b 3 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 y=1/2 · 8x •512=b3 •b=8 a=1/2 4.7 Assignment Page 285, 3-8 all 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 4.7 Write and Apply Exponential & Power Functions- Day 2 p. 509 How do you write a power function given two points? Which function uses logs to solve it? Modeling with POWER functions •y = b ax • Only 2 points are needed • (2,5) & (6,9) • 5 = a 2b • 9 = a 6b 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 Write a power function y = axb whose graph passes through (3, 2) and (6, 9) . SOLUTION STEP 1 Substitute the coordinates of the two given b points into y = ax . 2 = a 3b Substitute 2 for y and 3 for x. b 9=a 6 Substitute 9 for y and 6 for x. STEP 2 2 Solve for a in the first equation to obtain a = 3b , and substitute this expression for a in the second equation. 2 2 b Substitute 3b for a in 9 = 3b 6 second equation. 9=2 2b Simplify. 4.5 = b Divide each side by 2. Log2 2 4.5 = b Take log 2 of each side. Log 4.5 = b Change-of-base formula Log2 Use a calculator. 2.17 b STEP 3 2 Determine that a = 32.17 0.184. So, y = 0.184x 2.17 . b Write a power function y =ax whose graph passes through the given points. 6. (3, 4), (6, 15) SOLUTION STEP 1 Substitute the coordinates of the two given b points into y = ax . 4 = a 3b Substitute 4 for y and 3 for x. 15 = a 6 b Substitute 15 for y and 6 for x. STEP 2 4 Solve for a in the first equation to obtain a = 3b, and substitute this expression for a in the second equation. 4 4 b Substitute for a in 15 = 3b 6 3b second equation. b 15 = 4 2 Simplify. 15 = 2 b Divide each side by 4. 4 3.7 = 2 Log 2 3.7 = b Take log 2 of each side. Log 3.7 = b Log2 0.5682 0.3010 = 1.9 1.90 b Change-of-base formula Simplify. Use a calculator. STEP 3 4 Determine that a = 31.9 0.492. So, y = 0.492x 1.91 . Biology page 284 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. • 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 Homework • Page 285 15-20