Exponential Functions Dr. Laura J. Pyzdrowski

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1 1 Names: (4 communication points) About this Laboratory An exponential function is an example of a function that is not an algebraic combination of polynomials. Such functions are called trancendental functions. We will examine the graphs and such features as the domain and range of exponential functions of the form f(x) = ca x, where c, the coefficient, is a real number not equal to zero, and a, the base, is a real number greater than zero but not equal to one. In this laboratory, exact solutions for intercepts and asymptotes should be found algebraically and confirmed graphically. All answers in decimal form should be to the nearest tenth. Table entries should be expressed to the nearest thousandth. All intercepts should be written as points in the form (x, y) and an asymptote should be expressed using an equation. CP 1************************************************************************* First we will examine functions of the form f(x) = a x with base a greater than 1. Let f(x) = 2 x. Select Graph. (You may wish to zoom in to get a closer look around the x-axis.) Sketch f(x) = 2 x (1 point). For f(x) = 2 x, what is the y-intercept (2 points)? (a point) What is the horizontal asymptote (2 points)? (a line) What are the domain and range (4 points)? Clearly indicate each. (Be careful when using the computer to evaluate the domain and range.) When x = 0, f(x) = x = 1, f(x) =, and when (4 points). To arrive at your answers, analytically (by hand) substitute values into the equation.

2 2 ***************************************************************************** Construct a table in which x is an integer from -3 to 3 inclusive for f(x) = 2 x, 2f(x), 1/2f(x), and -2f(x). Your entries should be to the nearest thousandth. You may use the Grapher to calculate the table entries of one function at a time (2 points). k(x) = -2*2 x h(x) = ½*2 x g(x) = 2*2 x f(x) = 2 x x Notice that this is a special case example. When comparing f(x) to g(x) you see that the coefficient is the same as the base. We can also say that g(x) = 2 (x+1). (Make sure that you understand why.) What pattern do you see in the table that confirms the shift in the graphs from f(x) to g(x) (4 points)? In fact h(x) = 2 (x-1) and k(x) = -2 (x+1). Make sure that you understand why. Sketch each of the four functions on the same set of axes (1 point). (Be sure to identify each one on the sketch (8 points).) You may use the Grapher to find the graph of one function at a time. Choose a number other than 2 for a (the base) so that a > 1. Let f(x) = a x. Now explore graphs of g(x) = ca x by using different values of c. In other words, repeat the previous exercise with a new base and make notes in order to complete the Do later that follows.

3 3 Do later: Write a short paragraph explaining how, in general, f(x) = a x is affected by c to get the graph of g(x) = ca x. Be as complete as possible in your response to receive full points (8 points). CP 2************************************************************************* Let f(x) = e x. Select Graph. e. (2 points). (to three decimal places) Sketch f(x) = e x (1 point). What is the y-intercept (2 points)? What is the horizontal asymptote (2 points)? What are the domain and range (4 points)? (Clearly indicate each.) When x = 0, y = x = 1, y =, and when (4 points). Analytically (by hand) substitute values into the equation. Give exact values not approximations. ***************************************************************************** Sketch each of the following four functions on the same set of axes f(x) = e x, 2f(x), 1/2f(x), and -2f(x) (1 point). (Be sure to identify each one on the sketch (8 points).)

4 4 Complete the table (2 points). k(x) = -2*e x h(x) = ½*e x g(x) = 2*e x f(x) = e x x CP 3************************************************************************* Let f(x) = (1+1/x) x. Select Set function. Select the radio button next to x entries by hand. Enter increasing powers of 10 for the x values in the table. The f(x) values are approaching some real number. That number approximated to 3 decimals is: (2 points). As x 4, then f(x) (2 points). (As an irrational number.) (Symbolically this is asking what the y-values approach as the x-values go to infinity.) CP 4************************************************************************* Graph the functions f(x) = (1+1/x) x and g(x) = e to see a visualization of the concept of a limit. What happens to the points on the graph of f(x) as the x gets larger? (4 points). ***************************************************************************** Use the Same Computer Page and be sure to clear out the old graphs: So far we have examined two exponential functions of the form f(x) = ca x having the base a > 1, with the coefficient c understood to be 1": y = 2 x and y = e x. For each: 1. The domain has been (2 points). 2. The range has been (2 points). 3. When x = 1, y = (generalize) (2 points). 4. The y-intercept has been (2 points). 5. As x 4, then f(x) (2 points). 6. As x - 4, then f(x) (2 points). 7. These functions are both strictly increasing. This information is true for all exponential functions of the form f(x) = a x with a base greater than 1.

5 5 Explore on your own some exponential functions of the form f(x) = ca x with the base 0 < a < 1 and the coefficient of 1". Write a short paragraph describing the characteristics of these functions. Use the seven statements above to guide the content of your paragraph (14 points).

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