1. 4.7 Write and Apply Exponential & Power Functions
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. 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

6. 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 = x

7. 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

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. 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.

11. 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 = 2 3 b
9 = 2 2 b
Simplify.
4.5 = 2 b
Divide each side by 2.
Log = b 2
Take log of each side. 2
Log 4.5
Log2
= b
Change-of-base formula
Use a calculator. b
STEP 3
Determine that a = So, y = 0.184x
2.17 3 2

12. 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.

13. 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 =
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 = b 2
Take log of each side. 2

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

15. Biology page 284
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)?
• 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