Over Lesson 3-4 5–Minute Check 1 Solve 9 x – 2 = 27 3x. A. B.–1 C. D.

Over Lesson 3-4 5–Minute Check 1 Solve 9 x – 2 = 27 3x. A. B.–1 C. D.

Then/Now You modeled data using polynomial functions. (Lesson 2-1) Model data using exponential, logarithmic, and logistic functions. Linearize and analyze data.

Vocabulary logistic growth function linearize

Example 1 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.

Example 1 Exponential Regression Step 1Make 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.

Example 1 Exponential Regression Step 2Find 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 ≈ 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.

Example 1 Exponential Regression

Example 1 Exponential Regression Step 3Graph the regression equation and scatter plot on the same screen. Notice that the graph of the regression fits the data very well.

Example 1 Exponential Regression Step 4Use 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.

Example 1 Answer: B (t) = 400(1.19) t ; about 26,013 bacteria Exponential Regression

Example 1 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

Example 1 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

Example 2 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.

Example 2 Logarithmic Regression Step 1Let 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.

Example 2 Logarithmic Regression Step 2 Calculate the regression equation using LnReg. The correlation coefficient r ≈ – 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.

Example 2 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.

Example 2 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%

Example 2 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%

Example 2 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%

Key Concept 1

Example 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.

Example 3 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.

Example 3 Logistic Regression Step 2 Calculate the Logistic regression equation. Rounding values as shown yields the following logistic function for the data. P (x) =

Example 3 Logistic Regression Step 3 The graph of P (x) = fits the data very well as shown.

Example 3 Logistic Regression Step 4 The limit to growth in the modeling equation is the numerator of the fraction or 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%

Example 3 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%

Example 3 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%

Example 4 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.

Example 4 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.

Example 4 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 3The ExpReg equation does indeed fit the data very well as shown.

Example 4 Choose a Regression

Example 4 Answer:P (t) = 1000(1.78) t ; 7.99 years 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.

Example 4 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 B.PwrReg: y = x 0.05 C.ExpREg: y = 2000(1.02) x D.QuadReg: y = 0.42x x

Example 4 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 B.PwrReg: y = x 0.05 C.ExpREg: y = 2000(1.02) x D.QuadReg: y = 0.42x x

Key Concept 2

Example 5 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.

Example 5 Linearizing Data Step 1Graph a scatter plot of the data.

Example 5 Linearizing Data Step 2Linearize the data. To linearize the data that can be modeled by a power function, take the natural log of both x- and y-values.

Example 5 Linearizing Data Step 3Graph the linearized data and find the linear regression equation. The graph of (ln x, ln y) appears to cluster about a line. Let x = ln x and y = ln y. Using linear regression, the approximate equation modeling the linearized data is y = 3x ˆˆ ˆˆ

Example 5 Linearizing Data Step 4 Use the model for the linearized data to find a model for the original data. ˆˆ Replace x with ln x and y with ln y, and solve for y. ˆˆ y= 3x + 1.1Equation for linearized data e ln y = e 3 ln x Exponentiate each side y= e 3 ln x Inverse Property of Logarithms y= e 3ln x e 1.1 Product Property of Exponents y= e ln x 3 e 1.1 Power Property of Logarithms ln y= 3 ln x + 1.1x = ln x and y = ln y ˆˆ

Example 5 Linearizing Data y= x 3 e 1.1 Inverse Property of Logarithms y= 3x 3 e 1.1 ≈ 3 Therefore, a power function that models these data is y = 3x 3. Answer:y = 3x + 1; y = 3x 3 ˆˆ

Example 5 Assuming a power model, linearize the data to find the linear regression equation modeling the linearized data. A.y = 12x – 8.97 B.y = 17.07x + 6 C.y = 1.92x D.y = 1.2x – 0.27 ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

Example 5 Assuming a power model, linearize the data to find the linear regression equation modeling the linearized data. A.y = 12x – 8.97 B.y = 17.07x + 6 C.y = 1.92x D.y = 1.2x – 0.27 ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

Example 6 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

Example 6 Step 1Make 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

Example 6 Step 2 Graph the linearized data, and find a linear regression equation. Use Linearization A plot of the linearized data appears to form a straight line. Letting y = ln y, the regression equation is y = x ˆ ˆ

Example 6 Step 3Use the model for the linearized data to find a model for the original data. Use Linearization Replace y with ln y, and solve for y. ˆ y= x Equation for linearized data ˆ ln y= x y = ln y ˆ e ln y = e x Exponentiate each side. y= e x Inverse Property of Logarithms y= e x e 1.74 Product Property of Exponents y= 5.7e x e 1.74 ≈ 5.7

Example 6 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 = 5.7e x ; 125,551 bacteria Use Linearization

Example 6 The data in the table is modeled by a quadratic function. Linearize the data. A. B. C. D. 46

Example 6 The data in the table is modeled by a quadratic function. Linearize the data. A. B. C. D. 46

Vocabulary logistic growth function linearize