Transcript Section 7.7 PP.ppt

EXAMPLE 1
Write an exponential function
x
Write an exponential function y = ab whose graph
passes through (1, 12) and (3, 108).
SOLUTION
STEP 1
STEP 2
Substitute the coordinates of the two given
x
points into y = ab .
12 = ab 1
Substitute 12 for y and 1 for x.
108 = ab 3
Substitute 108 for y and 3 for x.
Solve for a in the first equation to obtain
12
a=
, and substitute this expression for
b
a in the second equation.
EXAMPLE 1
STEP 3
Write an exponential function
12 3
108 =
b b
Substitute 12 for a in second
equation. b
108 = 12b 2
Simplify.
9 = b2
Divide each side by 12.
3=b
Take the positive square root
because b > 0.
12 12
x
Determine that a = b =
=
4.
so,
y
=
4
3
.
3
EXAMPLE 2
Find an exponential model
Scooters A store sells motor scooters.
The table shows the number y of
scooters sold during the xth year that the
store has been open.
• Draw a scatter plot of the data pairs (x, ln y).
Is an exponential model a good fit for the
original data pairs (x, y)?
• Find an exponential model for the original data.
EXAMPLE 2
Find an exponential model
SOLUTION
STEP 1 Use a calculator to create a table of data
pairs (x, ln y).
x
ln y
1
2
3
4
5
6
7
2.48
2.77
3.22
3.58
3.91
4.29
4.56
STEP 2
Plot the new points as
shown. The points lie close
to a line, so an exponential
model should be a good fit
for the original data.
EXAMPLE 2
Find an exponential model
STEP 3
Find an exponential model y = ab x by choosing two
points on the line, such as (1, 2.48) and (7, 4.56). Use
these points to write an equation of the line. Then
solve for y.
ln y – 2.48 = 0.35(x – 1)
ln y = 0.35x + 2.13
Equation of line
Simplify.
y = e 0.35x + 2.13
Exponentiate each side using
base e.
y = e 2.13 (e0.35 ) x
y = 8.41(1.42) x
Use properties of exponents.
Exponential model
EXAMPLE 3
Use exponential regression
Scooters Use a graphing calculator to find an
exponential model for the data in Example 2. Predict
the number of scooters sold in the eighth year.
SOLUTION
Enter the original data into a
graphing calculator and perform
an exponential regression. The
model is y = 8.46(1.42) x .
Substituting x = 8 (for year 8) into the
model gives y = 8.46(1.42)8 140 scooters
sold.
GUIDED PRACTICE
for Examples 1, 2 and 3
x
Write an exponential function y = ab whose graph
passes through the given points.
1.
(1, 6), (3, 24)
SOLUTION
2.
3.
y=3 2
x
(2, 8), (3, 32)
SOLUTION
1 x
y= 2 4
(3, 8), (6, 64)
SOLUTION
y=2
x
GUIDED PRACTICE
4.
for Examples 1, 2 and 3
WHAT IF? In Examples 2 and 3, how would the
exponential models change if the scooter sales
were as shown in the table below?
SOLUTION
The initial amount would change to 11.39 and the
growth rate to 1.45.
EXAMPLE 4
Write a power function
Write a power function y = ax b whose graph passes
through (3, 2) and (6, 9) .
SOLUTION
STEP 1
Substitute the coordinates of the two given points
into y = ax b .
2 = a 3b
Substitute 2 for y and 3 for x.
9 = a 6b
Substitute 9 for y and 6 for x.
EXAMPLE 4
Write a power function
STEP 2
2
Solve for a in the first equation to obtain a = 3 b ,
and substitute this expression for a in the second
equation.
2
9 = 3b 6b
9 = 2 2b
4.5 = 2b
Log 2 4.5 = b
Log 4.5
Log2 = b
2.17 b
2
Substitute b for a in second
3
equation.
Simplify.
Divide each side by 2.
Take log 2 of each side.
Change-of-base formula
Use a calculator.
EXAMPLE 4
Write a power function
STEP 3
2
Determine that a = 32.17
0.184. So, y = 0.184x 2.17 .
GUIDED PRACTICE
for Example 4
Write a power function y = ax b whose graph passes
through the given points.
5.
(2, 1), (7, 6)
SOLUTION
6.
(3, 4), (6, 15)
SOLUTION
7.
y = 0.371x 1.43
y = 0.492x 1.91
(5, 8), (10, 34)
SOLUTION
y = 0.278x 2.09
GUIDED PRACTICE
8.
for Example 4
REASONING Try using the method of Example 4
to find a power function whose graph passes
through (3, 5) and (3, 7). What can you conclude?
SOLUTION
The points cannot form a power function.
EXAMPLE 5
Find a power model
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.
EXAMPLE 5
Find a power model
SOLUTION
STEP 1
Use a calculator to create a table of data pairs
(ln x, ln y).
STEP 2
Plot the new points as shown. The points lie close to
a line, so a power model should be a good fit for the
original data.
EXAMPLE 5
Find a power model
STEP 3
Find a power model y = axb by
choosing two points on the
line, such as (1.227, 0.525) and
(2.128, 2.774). Use these points
to write an equation of the line.
Then solve for y.
In y – y1 = m (In x – x 1)
Equation when axes are ln x
and ln y
In y – 2.774 = 2.5(In x – 2.128) Substitute.
In y = 2.5 In x – 2.546 Simplify.
In y = In x 2.5 – 2.546
Power property of logarithms
EXAMPLE 5
Writing Reciprocals
ln x 2.5 – 2.546
Y=e
Y=e
– 2.546
In x 2.5
e
Y = 0.0784x2.5
Exponentiate each side using
base e.
Product of powers property
Simplify.
EXAMPLE 6
Use power regression
Biology Use a graphing calculator to find a power
model for the data in Example 5. Estimate the weight
of a bird with a wingspan of 4.5 feet.
SOLUTION
Enter the original data into a graphing calculator
and perform a power regression. The model is
y = 0.0442x 2.87.
Substituting x = 4.5 into the model gives
2.87
y = 0.0442(4.5)
3.31 pounds.
GUIDED PRACTICE
9.
for Example 5 and 6
The table below shows the atomic number x and
the melting point y (in degrees Celsius) for the
alkali metals. Find a power model for the data.
SOLUTION
y = 397.61x-0.639