Transformations of Exponential Functions

Transcription

1 7-2 Transformations of Exponential Functions PearsonTEXAS.com SOLVE IT! f and g are exponential functions with the same base. Is the graph of g a compression, a reflection, or a translation of the graph of f? Or is it none of the above? Justify your reasoning. y f g Interactive Exploration Vocabulary Online O x Connect Mathematical Ideas (1)(F) How does this problem relate to a problem you have seen before? 227

2 Problem 1 Analyzing y = af (x) for f (x) = 2 x TEKS Process Standard (1)(D) Graph each function on the same set of axes as the parent function f (x) = 2 x. a. y = 0.5 # 2 x b. y =-4 # 2 x ELPS E P Discuss what you already know about stretching, compressing, and reflecting graphs with a classmate. What transformative effect does a in y = ab x have on the parent graph y = b x? Work together to create a model showing the effects of each case: a 6-1, -1 6 a 6 0, 0 6 a 6 1, and a 7 1. Problem 2 Analyzing y = af (x) for f (x) = 10 x TEKS Process Standard (1)(D) Graph each function on the same set of axes as the parent function f (x) = 10 x. What is the effect of each transformation on the range? a. y = 1 # x b. y =-10 x 228

3 Problem 3 Analyzing y = f (x) + d for f (x) = 2 x Graph each function on the same set of axes as the parent function f (x) = 2 x. What is the effect of each transformation on the y-intercept? a. y = 2 x - 5 b. y = 2 x + 1 Problem 4 Analyzing y = f (x) + d for f (x) = 10 x TEKS Process Standard (1)(D) Graph each function on the same set of axes as the parent function f (x) = 10 x. What is the effect of each transformation on the asymptote? a. y = 10 x - 10 b. y = 10 x

4 Problem 5 Analyzing y = f (x c) for f (x) = 2 x Graph each function on the same set of axes as the parent function f (x) = 2 x. What is the effect of each transformation on the domain? a. y = 2 (x+3) b. y = 2 (x-6) Problem 6 Analyzing y = f (x c) for f (x) = 10 x Graph each function on the same set of axes as the parent function f (x) = 10 x. What is the effect of each transformation on the asymptote? a. y = 10 (x-1) b. y = 10 (x+4) 230

5 TEKS Process Standard (1)(C) Problem 7 Using an Exponential Model a. The best temperature to brew coffee is between 195 F and 205 F. The table shows temperature readings from a sample cup of coffee. How long does it take for the coffee to reach a temperature of 100 degrees? Use an exponential model. Time (min) Temp ( F) b. In Problem 7, would the model of the exponential data be useful if you did not translate the data by 68 units? Explain. 231

6 Lesson Check Do you know HOW? 1. Graph the function y = 2 (x-1.5) on the same set of axes as the parent function. Then use transformations to describe how the two graphs are related. Math Tools Online Practice Virtual Nerd Tutorials 2. Write a function rule in the form y = 10 (x-c) + d for the function shown in the graph. 160 y (2, 110) 4 40 (0, 11) 2 (1, 20) O 2 4 x 3. The table shows the temperature of a cup of coffee as it cools. Given a room temperature of 71 F, how long does it take the coffee to cool from 100 F to 90 F? Use an exponential model and round to the nearest minute. Time (min) Temp ( F)

7 Lesson Check Do you UNDERSTAND? 4. Vocabulary Why are there multiple exponential parent functions rather than a single exponential parent function? Math Tools Online Practice Virtual Nerd Tutorials 5. Explain Mathematical Ideas (1)(G) A classmate was asked to graph y = 2 x - 2 and y = 2 x - 3. After making the graphs, your classmate says that the two graphs intersect. Do you think your classmate is correct? Explain. 6. Analyze Mathematical Relationships (1)(F) Write three different exponential functions that have the parent function y = 10 x and whose graphs pass through the point (0, 2). 233

8 TEXAS Test Practice Multiple Choice For Exercises 1 5, choose the correct letter. 1. The function y = 2 x + 3 models the number of people y who watch an online video each month, where x is the number of months since the video was created. Which of the following is a true statement about the graph of the function? A. The graph is a vertical stretch of the graph of the parent function y = 2 x. B. The graph intersects the line y = 3. C. The y-intercept of the graph is (0, 3). D. The graph intersects the line x = Which of the following functions has a graph that intersects the x-axis? F. y = 10 x - 2 H. y =-10 x - 2 G. y = 2 # 10 x J. y = 10 x A student used software to graph the function y = 2 x. Then he used the software to reflect the graph in the x-axis and shift it 2 units right. Which function matches the resulting graph? A. y =-2 x - 2 C. y =-2 (x-2) B. y =-2 x + 2 D. y =-2 (x+2) 4. Which transformation can be applied to the graph of y = 10 (x-4) to create the graph of y = 10 (x+3)? F. translation 7 units right H. translation 7 units down G. translation 7 units left J. translation 7 units up 5. Which function has a graph with an asymptote of the form y = k, where k 6 0? A. y =-10 x - 5 C. y =-2 (x-1) B. y = 10 (x-4) D. y = 2 x + 3 Short Response 6. Graph the functions y = 2 x and y =-2 # 2 x and use transformations to describe how the graphs are related. 234