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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?.

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4 7 write and apply exponential power functions

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?


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 ab x whose graph goes thru 1 6 3 24
Write an Exponential function, y=ab exponential curve.x 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 ab x whose graph goes thru 1 6 3 24 continued
Write an Exponential function, y=ab exponential curve.x 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


x exponential curve.

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 exponential curve.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


Write an exponential function y ab x whose graph goes thru 1 0625 2 32
Write an Exponential function, y=ab exponential curve.x 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


Power function
Power Function exponential curve.

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


Modeling with power functions
Modeling with POWER functions exponential curve.

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 exponential curve.

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 exponential curve.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 exponential curve.

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 exponential curve.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 exponential curve.

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 page 284

Biology exponential curve.

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.



Homework
Homework exponential curve.

  • Page 285

  • 3-7, 15-19


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