7.7 Write and Apply Exponential & Power Functions 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 24=6b2 4=b2 2=b a= 6/b = 6/2 = 3 So the function is Y=3·2x

Substitute the coordinates of the two given points into y = ab . 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 Substitute 12 for y and 1 for x. 108 = ab 3 Substitute 108 for y and 3 for x. STEP 2  

Substitute for a in second equation. 12 b 108 = b3 12 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 = 3 = 4 so, y = 4 3 . x

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

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

How do you write a power function given two points? Which function uses logs to solve it?

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

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

Substitute for a in second equation. 2 3 9 = 6 2 3 9 = 2 2 b Simplify. 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 Substitute for a in second equation. 2 3 b 9 = 6 2 3 b 9 = 2 2 b Simplify. 4.5 = 2 b Divide each side by 2. Log 4.5 = b 2 Take log of each side. 2 Log 4.5 Log2 = b Change-of-base formula Use a calculator. 2.17 b STEP 3 Determine that a = 0.184. So, y = 0.184x . 2.17 3 2

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

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

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

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