Activity 4.3 Exponential Graphs and Concavity Overview: This activity asks students to match exponential equations with their graphs. Students are then asked to describe characteristics of an exponential function given specific attributes of the function. Estimated Time Required: This activity should take approximately 15 minutes. Technology: None Concepts: The possible appearances of the graphs of exponential function The effect of the initial value on the appearance of the graph of an exponential function The effect of the growth factor on the appearance of the graph of an exponential function Asymptotes of exponential function The relationship between the behavior of the rate of change of a function and the concavity of the function’s graph. Discussion: This activity is an opportunity to build on students’ understanding of connection between the algebraic formula for an exponential function and the appearance of the graph. Make sure that the students appreciate that if the growth factor is between zero and one, then substituting large values of x into the formula is like multiplying by the growth factor over and over again. The net result of multiplying by a number 0 < b < 1 is that you get closer and closer to zero, which is why the height of the exponential function gets closer and closer to zero as x increases. Remind student how to test a table of data to check whether or not an exponential function could represent the given data. This can also serve as a second chance to illustrate the calculators for finding the formula for an exponential function. y 70 60 50 40 30 20 10 -4 -2 2 x 4 1. The graphs of the following functions are shown above. Label each graph with the letter of its corresponding function. a.) f x 2 0 1.4 d.) k x 10 0.8 x b.) g x 28 1.2 x c.) h x 15 0.6 x x 2. If an exponential function always has the form Q t abt , where a is the initial value, and b is the growth factor, can you make statements about the following: What is always true when the growth factor, b, is between 0 and 1? What is always true when the growth factor, b, is greater than 1? What occurs when the value of a is changed? What is the domain and range for these functions? Do these functions appear to have any asymptotes? Describe the rate of change of these functions.