Transcript File

Warm-Up Exercises
1. Write an equation in slope-intercept form for the line
through (2, 5) and (6, –3).
ANSWER
y = –2x + 9
2. Write an equation in point-slope form for the line
through (2, 4.53) and (5, 5.22).
ANSWER
y – 4.53 = 0.23(x – 2) or y – 5.22 = 0.23(x – 5)
Warm-Up Exercises
3. What is the value of y if the point (10, y) is on the line
y = 5.8x + 2.4?
ANSWER
60.4
Warm-Up1Exercises
EXAMPLE
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.
Warm-Up1Exercises
EXAMPLE
Write an exponential function
STEP 3
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
Warm-Up2Exercises
EXAMPLE
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.
Warm-Up2Exercises
EXAMPLE
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.
Warm-Up2Exercises
EXAMPLE
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
Warm-Up3Exercises
EXAMPLE
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.
Warm-Up
YOU
TRY Exercises
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
STEP 1
Substitute the coordinates of the two given
x
points into y = ab .
6 = ab1
Substitute 6 for y and 1 for x.
24 = ab 3
Substitute 24 for y and 3 for x.
Warm-Up
YOU
TRY Exercises
STEP 2
Solve for a in the first equation to obtain
6
a=
, and substitute this expression for
b
a in the second equation.
24 =
6
b
24 = 6b2
STEP 3
for Examples 1, 2 and 3
b3
Substitute 6 for a in second
equation. b
Simplify.
4 = b2
Divide each side by 6.
2=b
Take the positive square root
because b > 0.
6
6
x
Determine that a = b =
=
3.
so,
y
=
3
2
.
2
Warm-Up
YOU
TRY Exercises
for Examples 1, 2 and 3
x
Write an exponential function y = ab whose graph
passes through the given points.
2.
(2, 8), (3, 32)
SOLUTION
STEP 1
Substitute the coordinates of the two given
x
points into y = ab .
8 = ab1
Substitute 8 for y and 2 for x.
32 = ab 3
Substitute 32 for y and 3 for x.
Warm-Up
YOU
TRY Exercises
STEP 2
Solve for a in the first equation to obtain
8
a=
, and substitute this expression for
b2
a in the second equation.
32 =
32 = 8b
4=b
STEP 3
for Examples 1, 2 and 3
8
b2
b3
Substitute 8 for a in second
equation. b2
Divide each side by 4.
Take the positive square root
because b > 0.
8
8
1 .
Determine that a = 2 = 2 = 8 =
b
4 16 2
1 x
So, y = 2 4
Warm-Up
YOU
TRY Exercises
for Examples 1, 2 and 3
x
Write an exponential function y = ab whose graph
passes through the given points.
3.
(3, 8), (6, 64)
SOLUTION
STEP 1
Substitute the coordinates of the two given
x
points into y = ab .
8 = ab3
Substitute 8 for y and 3 for x.
64 = ab 6
Substitute 64 for y and 6 for x.
Warm-Up
YOU
TRY Exercises
STEP 2
for Examples 1, 2 and 3
Solve for a in the first equation to obtain
8
a = 3 , and substitute this expression for
b
a in the second equation.
64 =
8
b3
64= 8b3
64
3
b = 8 =8
b=2
b6
Substitute 83 for a in second
equation. b
Divide each side by 8.
Simplify.
Take the positive square root
because b > 0.
Warm-Up
YOU
TRY Exercises
STEP 3
for Examples 1, 2 and 3
8
8
Determine that a = b3 = 3 = 8 = 1 .
2
8
x
x
So, b = 2, a = 1, y = 1 2 = 2 .
Warm-Up
YOU
TRY Exercises
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.
Warm-Up4Exercises
EXAMPLE
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.
Warm-Up4Exercises
EXAMPLE
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.
Warm-Up4Exercises
EXAMPLE
Write a power function
STEP 3
2
Determine that a = 32.17
0.184. So, y = 0.184x 2.17 .
Warm-Up
YOU
TRY Exercises
for Example 4
Write a power function y = ax b whose graph passes
through the given points.
5.
(2, 1), (7, 6)
Warm-Up
YOU
TRY Exercises
for Example 4
Write a power function y = ax b whose graph passes
through the given points.
6.
(3, 4), (6, 15)
SOLUTION
STEP 1
Substitute the coordinates of the two given points
into y = ax b .
4 = a 3b
Substitute 4 for y and 3 for x.
15 = a 6b
Substitute 15 for y and 6 for x.
Warm-Up
YOU
TRY Exercises
for Example 4
STEP 2
4
Solve for a in the first equation to obtain a = 3 b ,
and substitute this expression for a in the second
equation.
4
15 = 3b 6b
15 = 4 2b
15
= 2b
4
3.7 = 2
Log 2 3.7 = b
4
Substitute b for a in second
3
equation.
Simplify.
Divide each side by 4.
Take log 2 of each side.
Warm-Up
YOU
TRY Exercises
Log 3.7
Log2 = b
0.5682
0.3010 = 1.9
1.90 b
for Example 4
Change-of-base formula
Simplify.
Use a calculator.
STEP 3
4
Determine that a = 31.9
0.492. So, y = 0.492x 1.91 .
Warm-Up
YOU
TRY Exercises
for Example 4
Write a power function y = ax b whose graph passes
through the given points.
7.
(5, 8), (10, 34)
SOLUTION
STEP 1
Substitute the coordinates of the two given points
into y = ax b .
8 = a 5b
Substitute 8 for y and 5 for x.
34 = a 10b
Substitute 34 for y and 10 for x.
Warm-Up
YOU
TRY Exercises
for Example 4
STEP 2
8
Solve for a in the first equation to obtain a = 5 b ,
and substitute this expression for a in the second
equation.
8
34 = 5b 10b
34 = 8 2b
17
= 2b
4
4.2 = 2
Log 2 4.2 = b
8
Substitute b for a in second
5
equation.
Simplify.
Take log 2 of each side.
Warm-Up
YOU
TRY Exercises
Log 4.2
Log2 = b
0.6284
0.3010 = b
2.09 b
for Example 4
Change-of-base formula
Simplify.
Use a calculator.
STEP 3
Determine that a =0.278. So, y = 0.278x
2.09
.
Warm-Up
YOU
TRY Exercises
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.
Warm-Up5Exercises
EXAMPLE
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.
Warm-Up5Exercises
EXAMPLE
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.
Warm-Up5Exercises
EXAMPLE
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 ( ln x – x1 )
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
Warm-Up5Exercises
EXAMPLE
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.
Warm-Up6Exercises
EXAMPLE
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.
Warm-Up
YOU
TRY Exercises
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 .
Warm-Up
Exercises
KEEP
GOING
1. Write an exponential function y = abx Whose graph
passes through (2, 48) and (4, 768).
ANSWERS
y = 3•4x
2. Find an exponential model for the data in the table.
ANSWERS
y = 19.4(1.81)x
Warm-Up
Exercises
KEEP
GOING
3. Write a power function y = axb whose graph
passes through (3, 8) and (6, 35).
ANSWERS
y = 0.77x2.13
4. Find a power model for the data in the table.
ANSWERS
y = 0.08x2.3