Name______________________________________________________________________________Date_________________Period______ Investigation: Comparing Power Functions In this investigation, we examine power functions of the form 𝑦 = 𝑘𝑥 ! , where k and r are real numbers. For most of the investigation, we let k=1 so we can focus on how r changes the behavior of the function. PART ONE DIRECTIONS: As you consider the following questions, you will need to adjust the window settings on your calculator. In the boxes provided, sketch two versions of the graphs as seen on your calculator. Label the x-­‐min, x-­‐
max, y-­‐min, and y-­‐max in your sketches. Color-­‐code or label the different functions so you can tell them apart. 1. In the same window, graph 𝑦 = 𝑥 ! , 𝑦 = 𝑥 ! , and 𝑦 = 𝑥 ! . What points do they have in common? For what values of x is 𝑥 ! > 𝑥 ! ? For what values of x is 𝑥 ! < 𝑥 ! ? What is the end behavior of each? Make a conjecture about the characteristics of the graph of 𝑦 = 𝑥 !! for any natural number n. 2. In the same window, graph 𝑦 = 𝑥 ! , 𝑦 = 𝑥 ! , and 𝑦 = 𝑥 ! . What points do they have in common? For what values of x is 𝑥 ! > 𝑥 ! ? For what values of x is 𝑥 ! < 𝑥 ! ? What is the end behavior of each? Make a conjecture about the characteristics of the graph of 𝑦 = 𝑥 !!!! for any natural number n. Adapted from December 14, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 3. In the same window, graph 𝑦 = 𝑥 !! and 𝑦 = 𝑥 !! . What points do they have in common? For what values of x is 𝑥 !! > 𝑥 !! ? For what values of x is 𝑥 !! < 𝑥 !! ? Are there any asymptotes? Where? What is the end behavior of each? Make a conjecture about the characteristics of the graph of 𝑦 = 𝑥 !!! for any natural number n. Make a conjecture about the characteristics of the graph of 𝑦 = 𝑥 !(!!!!) for any natural number n. 4. In the same window, graph 𝑦 = 𝑥 !/! and 𝑦 = 𝑥 !/! . What is the domain for each function? (The domain is different for each.) What is the range for each function? (The range is different for each.) What is the end behavior of each? Make a conjecture about the characteristics of the graph of 𝑦 = 𝑥 !/!! for any natural number n. Make a conjecture about the characteristics of the graph of 𝑦 = 𝑥 !/(!!!!) for any natural number n. Adapted from December 14, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. PART TWO DIRECTIONS: Use graphs and tables to help you answer the following questions. In the boxes provided, sketch the graphs as seen on your calculator. Label the x-­‐min, x-­‐max, y-­‐min, and y-­‐max in your sketches. Color-­‐code or label the different functions so you can tell them apart. 5. Order from least to greatest for 0 < 𝑥 < 1: 𝑦 = 𝑥 !/! , 𝑦 = 𝑥 ! , 𝑦 = 𝑥 ! , 𝑦 = 𝑥 !/! , 𝑦 = 𝑥 ! , and 𝑦 = 𝑥. 6. Order from least to greatest for 𝑥 > 1: 𝑦 = 𝑥 !/! , 𝑦 = 𝑥 ! , 𝑦 = 𝑥 ! , 𝑦 = 𝑥 !/! , 𝑦 = 𝑥 ! , and 𝑦 = 𝑥. 7. Order from least to greatest for 0 < 𝑥 < 1: 𝑦 = 𝑥 !! , 𝑦 = 𝑥 !! , 𝑦 = 𝑥 !! , and 𝑦 = 𝑥. Adapted from December 14, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 8. Order from least to greatest for 𝑥 > 1: 𝑦 = 𝑥 !! , 𝑦 = 𝑥 !! , 𝑦 = 𝑥 !! , and 𝑦 = 𝑥. 9. Match the functions to their graphs: a. 𝑦 = 𝑘𝑥 !/!" b. 𝑦 = 𝑘𝑥 !/!! k c. 𝑦 = 𝑘𝑥 !/! d. 𝑦 = 𝑘𝑥 !/! 1 Adapted from December 14, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.