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4.7 Write and Apply Exponential & Power Functions

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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?

- Substitute the coordinates into y=abx to get 2 equations.
- 1. 6=ab1
- 2. 24=ab3
- Then solve the system:

- 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

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

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

- .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

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

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

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.

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

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.

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

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

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

- Page 285
- 3-7, 15-19