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Exponential Function Investigation Go to the site http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=4 where you will find an applet which allows you to explore the graphs of exponential functions. Explore the graphs of exponential functions of the form y = ax by setting the 'M' and 'k' sliders to 1; (M=1) and (k=1). Set the ‘a’ slider to 1 (a=1). 1. The resulting graph is that of y = 1. What type of graph is this? 2. Why is the graph of y = 1 not that of an exponential function? 3. For a function of the form y = ax to be exponential, what are the restrictions on ‘a’? In other words, what value(s) can ‘a’ not have? Grab the ‘a’ slider and slide it left and right. 1. What is the y-intercept of the graph? Why? (Hint: Recall that x=0 when the graph crosses the y-axis.) 2. Does the graph ever cross the x-axis? In other words, does the graph have an x-intercept? Why or why not? (Hint: Keep in mind that when a graph crosses the x-axis, y = 0 giving the equation 0 = ax.) 3. What two quadrants does the graph never enter? What does this suggest about the range (values of y)? Why? To help you explain why, do the following: x y = ax Use your calculator to complete the given table of values for the exponential function y = 2x. 3 23 = In the table at the left, what do you notice about the value of y? 0.5 20.5 = 0 20 = -0.5 2-0.5 = -3 2-3 = -5 2-5 = Adjust the ‘a’ slider but restrict its setting to values greater than 1. 1. As the value of x gets larger and larger, what happens to the value of y? 2. As the value x gets smaller and smaller (closer to negative infinity), what value does y approach? This would suggest that a horizontal asymptote occurs at y = ____. (Hint: Recall your study of limits of infinity in lesson 6 of Polynomial and Rational Function Unit) Adjust the ‘a’ slider but restrict its setting to positive values less than 1. 1. As the value of x gets smaller and smaller, what happens to the value of y? 2. As the value x gets larger and larger (closer to positive infinity), what value does y approach. This would suggest that a horizontal asymptote occurs at y = ____. (Hint: Recall your study of limits of infinity in lesson 6 of Polynomial and Rational Function Unit) Conclusion The graph of an exponential function of the form y = ax, where a>0 and a ____, is a smooth curve with a y-intercept of _____ and a horizontal asymptote of y = _____. The functions domain is _______________ and its range is ____________________________________________________.

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exponential functions, exponential function, graphing calculator, exponential growth, table of values, graph of a function, how to, horizontal asymptote, calculator investigation, quadratic function, graph paper, sketch graphs, polynomial functions, quadratic equation, exponential and logarithmic functions

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views: | 29 |

posted: | 9/28/2010 |

language: | English |

pages: | 2 |

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