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Unit 4 MCR3U1 Lesson 4: Exploring the Properties of Exponential Functions Using a graphing calculator, graph each of the following exponential functions and complete the table below. Function Sketch x-intercept / y-intercept Asymptote Domain / Range y 2x 1 y 2 x y 2x 1 y 2x 1 x 1 y 1 2 Page 1 Unit 4 MCR3U1 Function Sketch x-intercept / y-intercept Asymptote Domain / Range x 1 y 1 2 y 2 x 1 y 2 x y 2 x 1 y 2 x1 1 Page 2 Unit 4 MCR3U1 Graphing Exponential Functions The graph of y b x can be illustrated on a graph. For example, we can graph the function y 2x (b = 2) using a table of values. x y -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 0.13 0.18 0.25 0.35 0.50 0.71 1.00 1.41 2.00 2.83 4.00 5.668 8.00 9 8 y 2x 7 6 5 4 3 2 1 0 -4 -2 0 2 4 Exponential functions have the same properties as linear and quadratic functions. These properties can be illustrated as follows. y-intercept: Substitute x = 0 into y 2 x . y 20 1 Therefore, the y-intercept is y = 1 x-intercept: Substitute y = 0 into y 2 x . 0 2x This equation has no real solutions since 2 x 0 for all real values of x. So, there is no x-intercept. Domain Since we can define 2x for all real values of x, the domain is x R. Range Since we can find a value of x for all positive real values of y, the range is y 0. Asymptotes Recall that an asymptote is a line that a curve approaches more and more closely but never touches. The asymptote for the graph of y 2 x is the line y = 0, or the x-axis. Page 3 Unit 4 MCR3U1 How would the graph change if b x -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 1 ? 2 9 y 8 5.66 4 2.83 2 1.41 1 0.71 0.5 0.35 0.25 0.18 0.13 8 7 6 5 4 1 y 2 x 3 2 1 0 -4 -2 2 0 4 The y-intercept for both graphs is y = 1. The domain and range of both graphs are the same. The only difference between the two graphs is that when b 1 , the graph of y b x is increasing. When 0 b 1,the graph of y b x is decreasing. How would the graph change if a 1.5 ? x -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 4 y 0.296 0.363 0.444 0.544 0.667 0.816 1.00 1.225 1.50 1.837 2.25 2.756 3.375 -4 The graph is increasing. However, it is not as close to the y-axis, so it is increasing slower. 4 3 3 2 2 1 1 0 -2 0 2 4 Page 4