Transcript y - Cloudfront.net

Over Lesson 3-4
Solve 9x – 2 = 273x.
A.
B. –1
C.
D.
Over Lesson 3-4
Solve 9x – 2 = 273x.
A.
B. –1
C.
D.
Over Lesson 3-4
Solve 3 + 5 log (2x) = 8.
A. 5
B.
C. 20
D. –2
Over Lesson 3-4
Solve 3e5x = 74. Round to the nearest hundredth.
A. 0.28
B. 0.64
C. 1.08
D. 0.47
Over Lesson 3-4
Solve 3e5x = 74. Round to the nearest hundredth.
A. 0.28
B. 0.64
C. 1.08
D. 0.47
Over Lesson 3-4
Solve 32x – 7 = 54x + 2. Round to the nearest
hundredth.
A. –4.5
B. –2.57
C. –0.39
D. no solution
Over Lesson 3-4
Solve 32x – 7 = 54x + 2. Round to the nearest
hundredth.
A. –4.5
B. –2.57
C. –0.39
D. no solution
Over Lesson 3-4
Solve ln (2x − 1) + ln (x + 3) = ln 9.
A.
B.
C.
D. no solution
Over Lesson 3-4
Solve ln (2x − 1) + ln (x + 3) = ln 9.
A.
B.
C.
D. no solution
Exponential Regression
BACTERIA The growth of a culture of bacteria is
shown in the table. Use exponential regression to
model the data. Then use your model to predict
how many bacteria there will be after 24 hours.
Exponential Regression
Step 1 Make a scatter plot.
Let B(t) represent the bacteria population
after t hours. Enter and graph the data on a
graphing calculator to create the scatter plot.
Notice that the plot very closely resembles
the graph of an exponential growth function.
Exponential Regression
Step 2 Find an exponential function to model
the data.
The bacteria population at time t = 0 is
represented by a and the growth rate,
18.9% per hour, is represented by b. Notice
that the correlation coefficient r ≈ 0.9999 is
close to 1, indicating a close fit to the data. In
the Y = menu, pick up this regression
equation by entering VARS , Statistics, EQ,
RegEQ.
Exponential Regression
Step 3 Graph the regression equation and scatter
plot on the same screen.
Notice that the graph of the regression fits the
data very well.
Exponential Regression
Step 4 Use the model to make a prediction.
The regression model is B(t) = 400(1.19)t. To
predict the bacteria population after 24 hours,
use the CALC feature to evaluate the function
for B(24) as shown. Based on the model, the
bacteria population will be about 26,013 after
24 hours.
Exponential Regression
Answer: B(t) = 400(1.19)t; about 26,013 bacteria
The number of leaves falling per hour from the
trees in an arboretum is shown in the table below.
Use an exponential regression model to predict
how many leaves will fall in the tenth hour.
A. 415
B. 500
C. 485
D. 622
The number of leaves falling per hour from the
trees in an arboretum is shown in the table below.
Use an exponential regression model to predict
how many leaves will fall in the tenth hour.
A. 415
B. 500
C. 485
D. 622
Logarithmic Regression
MEMORY A group of students studied a
photograph for 30 seconds. Beginning 1 day later,
a test was given each day to test their memory of
the photograph. The average score for each day is
shown in the table. Use logarithmic regression to
model the data. Then use your model to predict the
average test score after 2 weeks.
Logarithmic Regression
Step 1
Let S(t) represent the students’ average
score t days after studying the photograph.
Enter and graph the data on a graphing
calculator to create the scatter plot.
Logarithmic Regression
Step 2
Calculate the regression equation using
LnReg. The correlation coefficient
r ≈ –0.9835 indicates a close fit to the data.
Rounding each value to two decimal places,
a natural logarithm function that models the
data is S(t) = 73.1 – 7.31ln x.
Logarithmic Regression
Step 3
The figure shows the results of the
regression S(t). The graph of
S(t) = 73.1 – 7.31 ln x fits the data very well.
Logarithmic Regression
Step 4
Answer:
To predict the students’ average score after
14 days, use the CALC feature to evaluate
the function for S(14) as shown. Based on
the model, the students’ average score will
be about 53.8%.
Logarithmic Regression
Step 4
To predict the students’ average score after
14 days, use the CALC feature to evaluate
the function for S(14) as shown. Based on
the model, the students’ average score will
be about 53.8%.
Answer: S(t) = 73.1 – 7.31 ln t; 53.8%
MEMORY Students do not remember everything
presented to them in a mathematics class. The table
below shows the average percentage of information
retained t days after the lesson by a group of
students. Use a logarithmic regression model to
predict the students’ retention percentage on the
tenth day.
A. 46.3%
B. 45.9%
C. 45.1%
D. 43.3%
MEMORY Students do not remember everything
presented to them in a mathematics class. The table
below shows the average percentage of information
retained t days after the lesson by a group of
students. Use a logarithmic regression model to
predict the students’ retention percentage on the
tenth day.
A. 46.3%
B. 45.9%
C. 45.1%
D. 43.3%
Logistic Regression
ADVERTISING The number of television ads for a
certain product affects the percentage of people
who purchase the product as shown in the table.
Use logistic regression to find a logistic growth
function to model the data. Then use your model
to predict the limit to the percentage of people
who will purchase the product.
Logistic Regression
Step 1
Let P(x) represent the percentage of people
who purchase a product after x number of
ads. The scatter plot of the data resembles
the graph of a logistic growth function.
Logistic Regression
Step 2
Calculate the Logistic regression equation.
Rounding values as shown yields the
following logistic function for the data.
P(x) =
Logistic Regression
Step 3
The graph of P(x) =
data very well as shown.
fits the
Logistic Regression
Step 4
Answer:
The limit to growth in the modeling equation
is the numerator of the fraction or 55.86.
Therefore, according to this model, the
number of people who purchase an item
based on the number of adds will approach,
but will never reach 56%.
Logistic Regression
Step 4
The limit to growth in the modeling equation
is the numerator of the fraction or 55.86.
Therefore, according to this model, the
number of people who purchase an item
based on the number of adds will approach,
but will never reach 56%.
Answer: P(x) =
; about 56%
MUSIC The probability of a person liking a song
increases with the number of friends who say they
also like the song. The data shown in the table
models this situation. Use a logistic growth
function to determine the limit to the probability
that a person will like a song based on the number
of friends who say they like the song.
A. about 31%
B. about 27%
C. about 56%
D. about 29%
MUSIC The probability of a person liking a song
increases with the number of friends who say they
also like the song. The data shown in the table
models this situation. Use a logistic growth
function to determine the limit to the probability
that a person will like a song based on the number
of friends who say they like the song.
A. about 31%
B. about 27%
C. about 56%
D. about 29%
Choose a Regression
INTERNET Use the data in the table to determine a
regression equation that best relates the profit of a
Web site with the time it has been in business.
Then determine the approximate time it will take
for the Web site to earn a profit of $100,000 in one
year.
Choose a Regression
Step 1 From the shape of the scatter plot shown, it
appears that these data could best be
modeled by the exponential regression
model.
Choose a Regression
Step 2 Use the LinReg(ax + b), QuadReg,
CubicReg, LnReg, ExpReg, PwrReg, and
Logistic regression equations to fit the data,
noting the corresponding correlation
coefficients. The regression equation with a
correlation coefficient closest to 1 is the
ExpReg with equation rounded to
P(t) = 1000(1.78)t.
Step 3 The ExpReg equation does indeed fit the
data very well as shown.
Choose a Regression
Choose a Regression
Step 4 Use the CALC INTERSECT feature of your
calculator to find the value of t when
P(t) = 100,000. The intersection of the two
graphs, y = 100,000 and P(t) = 1000(1.78)t is
shown to be about 7.99 years.
Answer:
Choose a Regression
Step 4 Use the CALC INTERSECT feature of your
calculator to find the value of t when
P(t) = 100,000. The intersection of the two
graphs, y = 100,000 and P(t) = 1000(1.78)t is
shown to be about 7.99 years.
Answer: P(t) = 1000(1.78)t; 7.99 years
BUSINESS Use the data in the table to determine
a regression equation that best relates the profit of
a business with the time it has been in business.
A. LinReg(ax + b): y = 42.05x + 1997.11
B. PwrReg: y = 2027.16x 0.05
C. ExpREg: y = 2000(1.02)x
D. QuadReg: y = 0.42x 2 + 39.52x + 2000.06
BUSINESS Use the data in the table to determine
a regression equation that best relates the profit of
a business with the time it has been in business.
A. LinReg(ax + b): y = 42.05x + 1997.11
B. PwrReg: y = 2027.16x 0.05
C. ExpREg: y = 2000(1.02)x
D. QuadReg: y = 0.42x 2 + 39.52x + 2000.06
Linearizing Data
Make a scatter plot of the data, and linearize the
data assuming a power model. Graph the
linearized data, and find the linear regression
equation. Then use this linear model to find a
model for the original data.
Linearizing Data
Step 1
Graph a scatter plot of the data.
Linearizing Data
Step 2
Linearize the data.
To linearize the data that can be modeled by
a power function, take the natural log of
both x- and y-values.
Linearizing Data
Step 3
Graph the linearized data and find the
linear regression equation.
The graph of (ln x, ln y) appears to cluster
about a line. Let xˆ = ln x and yˆ = ln y. Using
linear regression, the approximate equation
modeling the linearized data is
yˆ = 3xˆ + 1.1.
Linearizing Data
Step 4
Use the model for the linearized data to
find a model for the original data.
Replace xˆ with ln x and yˆ with ln y, and
solve for y.
yˆ = 3xˆ + 1.1
Equation for linearized data
ln y = 3 ln x + 1.1
xˆ = ln x and yˆ = ln y
eln y = e3 ln x + 1.1
Exponentiate each side
y = e3 ln x + 1.1
Inverse Property of Logarithms
y = e3ln xe1.1
Product Property of Exponents
ln x
y =e
3
e1.1
Power Property of Logarithms
Linearizing Data
y = x 3e1.1
Inverse Property of Logarithms
y = 3x 3
e1.1 ≈ 3
Therefore, a power function that models these data is
y = 3x 3.
Answer:
Linearizing Data
y = x 3e1.1
Inverse Property of Logarithms
y = 3x 3
e1.1 ≈ 3
Therefore, a power function that models these data is
y = 3x 3.
Answer: yˆ = 3xˆ + 1; y = 3x 3
Assuming a power model, linearize the data to find
the linear regression equation modeling the
linearized data.
A. yˆ = 12xˆ – 8.97
B. yˆ = 17.07xˆ + 6
C. yˆ = 1.92xˆ + 1.11
D. yˆ = 1.2xˆ – 0.27
Assuming a power model, linearize the data to find
the linear regression equation modeling the
linearized data.
A. yˆ = 12xˆ – 8.97
B. yˆ = 17.07xˆ + 6
C. yˆ = 1.92xˆ + 1.11
D. yˆ = 1.2xˆ – 0.27
Use Linearization
BACTERIA The table shows the number of
bacteria found in a culture. Find an exponential
model relating these data by linearizing the data
and finding the linear regression equation. Then
use your model to predict the number of bacteria
after 10 hours.
Use Linearization
Step 1
Make a scatter plot and linearize the data.
The scatter plot shown is nonlinear and its
shape suggests that the data could be
modeled by an exponential function.
Linearize the data by finding (x, ln y).
Use Linearization
Step 2
Graph the linearized data, and find a
linear regression equation.
A plot of the linearized data appears to form
a straight line. Letting yˆ = ln y, the
regression equation is yˆ = x + 1.74.
Use Linearization
Step 3
Use the model for the linearized data to
find a model for the original data.
Replace yˆ with ln y, and solve for y.
yˆ = x + 1.74
Equation for linearized data
ln y = x + 1.74
yˆ = ln y
eln y = ex + 1.74
Exponentiate each side.
y = ex + 1.74
Inverse Property of Logarithms
y = exe1.74
Product Property of Exponents
y = 5.7ex
e1.74 ≈ 5.7
Use Linearization
Step 4
Use the equation that models the original
data to solve the problem.
To find the number of bacteria after 10
hours, find y when x = 10. According to this
model the number of bacteria after 10 hours
will be about 125,551 bacteria.
Answer:
Use Linearization
Step 4
Use the equation that models the original
data to solve the problem.
To find the number of bacteria after 10
hours, find y when x = 10. According to this
model the number of bacteria after 10 hours
will be about 125,551 bacteria.
Answer: y = 5.7ex; 125,551 bacteria
The data in the table is modeled by a
quadratic function. Linearize the data.
A.
C.
46
D.
B.
The data in the table is modeled by a
quadratic function. Linearize the data.
A.
C.
46
D.
B.