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MCR 3U0 Name: _____________________ In this activity, you will be exploring a specific kind of graph; the graph of exponential functions. This activity will involve the dropping of two different balls. You will be observing how their height changes with each bounce after they have fallen. When you drop a ball it will bounce several times. Is the height of each bounce related to the height of the previous bounce? Part A: Soccer ball 1) Complete the table below with your data for the soccer ball: Bounce Bounce First Bounce Bounce First Number Height Differences Number Height Differences 0 0 1 1 2 2 3 3 2) Plot the bounce height vs. the bounce number below for each table in two different colours. Draw a dashed curve through the set of points. Part B: Basketball 1) Repeat all steps above, however, this time, use a basketball. Bounce Bounce First Second Bounce Bounce First Second Number Height Differences Differences Number Height Differences Differences 0 0 1 1 2 2 3 3 2) Plot the bounce height vs. the bounce number below for each table in two different colours. Draw a dashed curve through the set of points. Reflection 1) Describe the shape of each graph. 2) Is the shape of the graph linear or quadratic? Give two reasons to justify your answer. 3) Describe how the bounce height changed from one bounce to the next. Was this pattern the same for each type of ball? 4) If you continue the pattern indefinitely, will the bounce height ever reach zero? 4.5 Exploring……… Purpose of Investigation: To determine the characteristics of the graphs and equations of exponential functions. x Functions such as f ( x) 2 & g ( x) are examples of exponential functions. 1 x 2 Exponential functions are used to model growth and decay. Investigation-Part A: 1. Complete the chart. Equations of Function Table of Values Domain Range Intercepts Asymptotes x y -3 y 2x -2 -1 0 1 2 3 4 x y -3 y 10x -2 -1 0 1 2 3 4 2. Graph the functions on the same grid. Draw a smooth curve through each set of points. Label each curve with the appropriate equation. 3. In each of the tables, calculate the first and second differences. Describe any patterns you see. How can you tell that a function is exponential from its differences? 4. For each function, describe how values of the dependent variable change as the values of the independent variable increase and decrease. 5. Which curve increases fastest as the values of x increases? __________________ 6. The base of an exponential function of the form y b cannot be 1. Explain why this x restriction is necessary. Investigation-Part B: 7. Complete the chart. Equations of Function Table of Values Domain Range Intercepts Asymptotes x y x -3 1 y -2 2 -1 0 1 2 3 4 x y -3 x 1 y -2 5 -1 0 1 2 3 4 x y -3 x 1 y -2 10 -1 0 1 2 3 4 8. Graph the functions on the same grid. Draw a smooth curve through each set of points. Label each curve with the appropriate equation. 9. For each function, describe how values of the dependent variable change as the values of the independent variable increase and decrease. 10. How do the graphs from Part B differ from those in Part A? Summary: 11. Compare the features of the graphs of y b x for each Part. Comment on the domain, range, intercepts, and asymptotes and a) different values of b when b 1 b) different values of b when 0 b 1 c) values of b when 0 b 1 , compared with values of b 1 d) values of b 0 compared with values of b 0 Extension: 12. Investigate what happens when the base of an exponential function is negative. Try y (2)x . Discuss your findings. 13. How do the differences for exponential functions differ from those for linear and quadratic functions?
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