Write and Apply Exponential and Power Functions

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TEKS 7.7 a., 2A..B, 2A..F Write and Apply Exponential and Power Functions Before You wrote linear, quadratic, and other polynomial functions. Now You will write exponential and power functions. Why? So you can model biology problems, as in Example 5. Key Vocabulary power function, In Chapter 2, you learned that two points determine a line. Similarly, two points determine an exponential curve. p. 428 exponential function, p. 478 E XAMPLE Write an exponential function Write an exponential function y 5 ab x whose graph passes through (, 2) and (, 08). STEP Substitute the coordinates of the two given points into y 5 ab x. 2 5 ab Substitute 2 for y and for x. 08 5 ab Substitute 08 for y and for x. STEP 2 Solve for a in the first equation to obtain a 5 2 } b, and substitute this expression for a in the second equation. 08 5 2 } b 2 b Substitute 2 } b for a in second equation. 08 5 2b 2 Simplify. 95 b 2 Divide each side by 2. 5 b Take the positive square root because b > 0. STEP Determine that a 5 2 } b 5 2 } 5 4. So, y 5 4 p x. TRANSFORMING EXPONENTIAL DATA A set of more than two points (x, y) fits an exponential pattern if and only if the set of transformed points (x, ln y) fits a linear pattern. Graph of points (x, y) Graph of points (x, ln y) y 5 2 x s22, 4d s2, 2d 2 y (0, ) (, 2) x ln y 5 x(ln 2) ln y (0, 0) (, 0.69) (2, 20.69) x (22, 2.9) The graph is an exponential curve. The graph is a line. 7.7 Write and Apply Exponential and Power Functions 529

E XAMPLE 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. Year, x 2 4 5 6 7 Number of scooters sold, y 2 6 25 6 50 67 96 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. STEP Use a calculator to create a table of data pairs (x, ln y). x 2 4 5 6 7 ln y 2.48 2.77.22.58.9 4.20 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. ln y (7, 4.56) USE POINT-SLOPE FORM Because the axes are x and ln y, the point-slope form is rewritten as ln y 2 y 5 m(x 2 x ). The slope of the line through (, 2.48) and (7, 4.56) is: 4.56 2 2.48 } ø 0.5 7 2 STEP Find an exponential model y 5 ab x by choosing two points on the line, such as (, 2.48) and (7, 4.56). Use these points to write an equation of the line. Then solve for y. ln y 2 2.48 5 0.5(x 2 ) ln y 5 0.5x 2. Equation of line Simplify. y5 e 0.5x 2. Exponentiate each side using base e. y5 e 2. (e 0.5 ) x Use properties of exponents. (, 2.48) x y5 8.4(.42) x Exponential model EXPONENTIAL REGRESSION A graphing calculator that performs exponential regression uses all of the original data to find the best-fitting model. E XAMPLE 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. Enter the original data into a graphing calculator and perform an exponential regression. The model is y 5 8.46(.42) x. Substituting x 5 8 (for year 8) into the model gives y 5 8.46(.42) 8 ø 40 scooters sold. ExpReg y=a*b^x a=8.4577797 b=.4884860 50 Chapter 7 Exponential and Logarithmic Functions

GUIDED PRACTICE for Examples, 2, and Write an exponential function y 5 ab x whose graph passes through the given points.. (, 6), (, 24) 2. (2, 8), (, 2). (, 8), (6, 64) 4. WHAT IF? In Examples 2 and, how would the exponential models change if the scooter sales were as shown in the table below? Year, x 2 4 5 6 7 Number of scooters sold, y 5 2 40 52 80 05 40 WRITING POWER FUNCTIONS Recall from Lesson 6. that a power function has the form y 5 ax b. Because there are only two constants (a and b), only two points are needed to determine a power curve through the points. E XAMPLE 4 Write a power function Write a power function y 5 ax b whose graph passes through (, 2) and (6, 9). STEP Substitute the coordinates of the two given points into y 5 ax b. 2 5 a p b Substitute 2 for y and for x. 9 5 a p 6 b Substitute 9 for y and 6 for x. STEP 2 Solve for a in the first equation to obtain a 5 } 2, and substitute this b expression for a in the second equation. 9 5 2 } b 2 6 b Substitute 2 } b 9 5 2 p 2 b Simplify. 4.5 5 2 b Divide each side by 2. for a in second equation. log 2 4.5 5 b Take log 2 of each side. log 4.5 } log 2 5 b Change-of-base formula 2.7 ø b Use a calculator. STEP Determine that a 5 2 } 2.7 ø 0.84. So, y 5 0.84x2.7. GUIDED PRACTICE for Example 4 Write a power function y 5 ax b whose graph passes through the given points. 5. (2, ), (7, 6) 6. (, 4), (6, 5) 7. (5, 8), (0, 4) 8. REASONING Try using the method of Example 4 to find a power function whose graph passes through (, 5) and (, 7). What can you conclude? 7.7 Write and Apply Exponential and Power Functions 5

TRANSFORMING POWER DATA A set of more than two points (x, y) fits a power pattern if and only if the set of transformed points (ln x, ln y) fits a linear pattern. Graph of points (x, y) Graph of points (ln x, ln y) y y 5 x /2 (,.7) (6, 2.45) (4, 2) ln y ln y 5 ln x 2 (.9, 0.69) (, ) x (.79, 0.9) (., 0.55) (0, 0) 2 ln x The graph is a power curve. The graph is a line. E XAMPLE 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. STEP Use a calculator to create a table of data pairs (ln x, ln y). ln x 0.642.072.227.677 2.28 ln y 2.470 0.09 0.525.9 2.774 USE POINT-SLOPE FORM The slope of the line is 2.774 2 0.525 } ø 2.50. 2.28 2.227 STEP 2 STEP 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. Find a power model y 5 ax b by choosing two points on the line, such as (.227, 0.525) and (2.28, 2.774). Use these points to write an equation of the line. Then solve for y. ln y 2 y 5 m(ln x 2 x ) ln y 2 2.774 5 2.5(ln x 2 2.28) ln y 5 2.5 ln x 2 2.546 ln y 5 ln x 2.5 2 2.546 y5 e ln x2.5 2 2.546 y5 e 22.546 ln x2.5 p e Product y5 0.0784x 2.5 Equation when axes are ln x and ln y Substitute. Simplify. Power property of logarithms Exponentiate each side using base e. of powers property Simplify. ln y (.227, 0.525) (2.28, 2.774) ln x 52 Chapter 7 Exponential and Logarithmic Functions

POWER REGRESSION A graphing calculator that performs power regression uses all of the original data to find the best-fitting model. E XAMPLE 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. Enter the original data into a graphing calculator and perform a power regression. The model is y 5 0.0442x 2.87. Substituting x 5 4.5 into the model gives y 5 0.0442(4.5) 2.87 ø. pounds. PwrReg y=a*x^b a=.044266 b=2.8777024 GUIDED PRACTICE for Examples 5 and 6 9. 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. Alkali metal Lithium Sodium Potassium Rubidium Cesium Atomic number, x 9 7 55 Melting point, y 80.5 97.8 6.7 8.9 28.5 7.7 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Exs., 2, and 5 TAKS PRACTICE AND REASONING Exs. 27,, 5, 7, and 8. VOCABULARY Copy and complete: Given a set of more than two data pairs (x, y), you can decide whether a(n)? function fits the data well by making a scatter plot of the points (x, ln y). 2. WRITING Explain how you can determine whether a power function is a good model for a set of data pairs (x, y). EXAMPLE on p. 529 for Exs. 0 EXAMPLE 2 on p. 50 for Exs. 4 WRITING EXPONENTIAL FUNCTIONS Write an exponential function y 5 ab x whose graph passes through the given points.. (, ), (2, 2) 4. (2, 24), (, 44) 5. (, ), (5, 4) 6. (, 27), (5, 24) 7. (, 2), (, 50) 8. (, 40), (, 640) 9. (2, 0), (4, 0.) 0. (2, 6.4), (5, 409.6) FINDING EXPONENTIAL MODELS Use the points (x, y) to draw a scatter plot of the points (x, ln y). Then find an exponential model for the data.. (, 8), (2, 6), (, 72), (4, 44), (5, 288) 2. (,.), (2, 0.), (, 0.6), (4, 92.7), (5, 280.9). (, 9.8), (2, 2.2), (, 5.2), (4, 9), (5, 2.8) 4. (,.4), (2, 6.7), (, 2.9), (4. 6.4), (5, 790.9) 7.7 Write and Apply Exponential and Power Functions 5

EXAMPLE 4 on p. 5 for Exs. 5 22 WRITING POWER FUNCTIONS Write a power function y 5 ax b whose graph passes through the given points. 5. (4, ), (8, 5) 6. (5, 9), (8, 4) 7. (2, ), (6, 2) 8. (, 4), (9, 44) 9. (4, 8), (8, 0) 20. (5, 0), (2, 8) 2. (4, 6.2), (7, 2) 22. (., 5), (6.8, 9.7) EXAMPLE 5 on p. 52 for Exs. 2 26 FINDING POWER MODELS Use the given points (x, y) to draw a scatter plot of the points (ln x, ln y). Then find a power model for the data. 2. (, 0.6), (2, 4.), (, 2.4), (4, 27), (5, 49.5) 24. (,.5), (2, 4.8), (, 9.5), (4, 5.4), (5, 22.) 25. (, 2.5), (2,.7), (, 4.7), (4, 5.5), (5, 6.2) 26. (, 0.8), (2, 0.99), (,.), (4,.2), (5,.29) 27. MULTIPLE TAKS REASONING CHOICE Which equation is equivalent to log y 5 2x? A y 5 0(00) x B y 5 0 x C y 5 e 2x D y 5 e 2 ERROR ANALYSIS Describe and correct the error in writing y as a function of x. 28. ln y 5 2x y 5 e 2x 29. ln y 5 ln x 2 2 ln y 5 ln x 2 2 ln x 2 2 y 5 e y 5 e 2x e y 5 (e 2 ) x e y 5 e ln x p e 22 y 5 7.9 x 2.72 y 5 (x)(0.5) 5 0.405x 0. CHALLENGE Take the natural logarithm of both sides of the equations y 5 ab x and y 5 ax b. What are the slope and y-intercept of the line relating x and ln y for y 5 ab x? of the line relating ln x and ln y for y 5 ax b? PROBLEM SOLVING GRAPHING CALCULATOR You may wish to use a graphing calculator to complete the following Problem Solving exercises. EXAMPLES 2,, 5, and 6 on pp. 50 5 for Exs. 5. BIOLOGY Scientists use the circumference of an animal s femur to estimate the animal s weight. The table shows the femur circumference C (in millimeters) and the weight W (in kilograms) for several animals. Animal Giraffe Polar bear Lion Squirrel Otter C (mm) 7 5 9.5 28 W (kg) 70 448 4 0.99 9.68 a. Draw a scatter plot of the data pairs (ln C, ln W). b. Find a power model for the original data. c. Predict the weight of a cheetah if the circumference of its femur is 68.7 millimeters. 5 WORKED-OUT SOLUTIONS 54 Chapter 7 Exponential p. WS and Logarithmic Functions 5 TAKS PRACTICE AND REASONING

2. ASTRONOMY The table shows the mean distance x from the sun (in astronomical units) and the period y (in years) of six planets. Draw a scatter plot of the data pairs (ln x, ln y). Find a power model for the original data. Planet Mercury Venus Earth Mars Jupiter Saturn x 0.87 0.72.000.524 5.20 9.59 y 0.24 0.65.000.88.862 29.458. SHORT TAKS REASONING RESPONSE The table shows the numbers of business and nonbusiness users of instant messaging for the years 998 2004. Years since 997 2 4 5 6 7 Business users (in millions) 2 5 7 20 40 80 Non-business users (in millions) 55 97 40 60 95 25 260 a. Find an exponential model for the number of business users over time. b. Explain how to tell whether a linear, exponential, or power function best models the number of non-business users over time. Then find the bestfitting model. 4. MULTI-STEP PROBLEM The boiling point of water increases with atmospheric pressure. At sea level, where the atmospheric pressure is about 760 millimeters of mercury, water boils at 008C. The table shows the boiling point T of water (in degrees Celsius) for several different values of atmospheric pressure P (in millimeters of mercury). a. Graph Draw a scatter plot of the data pairs (ln P, ln T). b. Model Find a power model for the original data. c. Predict When the atmospheric pressure is 620 millimeters of mercury, at what temperature does water boil? P T 49 60 24 70 55 80 526 90 760 00 075 0 5. EXTENDED TAKS REASONING RESPONSE Your visual near point is the closest point at which your eyes can see an object distinctly. Your near point moves farther away from you as you grow older. The diagram shows the near point y (in centimeters) at age x (in years). a. Graph Draw a scatter plot of the data pairs (x, ln y). b. Graph Draw a scatter plot of the data pairs (ln x, ln y). c. Interpret Based on your scatter plots, does an exponential function or a power function best fit the original data? Explain your reasoning. d. Model Based on your answer for part (c), write a model for the original data. Use your model to predict the near point for an 80-year-old person. 7.7 Write and Apply Exponential and Power Functions 55

6. CHALLENGE A doctor measures an astronaut s pulse rate y (in beats per minute) at various times x (in minutes) after the astronaut has finished exercising. The results are shown in the table. The astronaut s resting pulse rate is 70 beats per minute. Write an exponential model for the data. x 0 2 4 6 8 0 2 4 y 72 2 0 92 84 78 75 72 MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 4.; TAKS Workbook 7. TAKS PRACTICE A poster is 8 inches taller than it is wide. The area of the poster is 84 square inches. Which equation can be used to find the width of the poster? TAKS Obj. 0 A x 8 5 84 B x(x 8) 5 84 C x 2 (x 8) 2 5 84 2 D x (x 8) 5 84 x x 8 REVIEW Lesson 2.; TAKS Workbook 8. TAKS PRACTICE Use the graph of y 52} x to solve the 4 equation for x when y 522. TAKS Obj. 4 F x 5 G x 5 4 H x 5 8 J x 5 4 2 242222 2 4 x 22 2 24 y QUIZ for Lessons 7.6 7.7 Solve the equation. Check for extraneous solutions. (p. 55). 2 x 5 6 x 2 2. e 2x 5 4. 2x 5 5 4. x 2 5 5 0 5. log 4 (4x 7) 5 log 4 x 6. ln (x 2 2) 5 ln 6x 7. log x 52 8. 6 ln x 5 0 9. log 2 (x 4) 5 5 Write an exponential function y 5 ab x whose graph passes through the given points. (p. 529) 0. (, 5), (2, 0). (, 4), (2, 2) 2. (2, 5), (, 45) Write a power function y 5 ax b whose graph passes through the given points. (p. 529). (4, 8), (9, 2) 4. (, 2), (0, 6) 5. (5, 4), (, 5) 6. BIOLOGY The average weight y (in kilograms) of an Atlantic cod from the Gulf of Maine can be modeled by y 5 0.5(.46) x where x is the age of the cod (in years). Estimate the age of a cod that weighs 5 kilograms. (p. 55) 56 Chapter 7 EXTRA Exponential PRACTICE and Logarithmic for Lesson Functions 7.7, p. 06 ONLINE QUIZ at classzone.com

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