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Introduction

Introduction

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Introduction

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  1. Introduction In this lesson, different methods will be used to graph exponential functions and analyze the key features of the graph. In an exponentialfunction, the graph is a smooth line with a rounded curve. All exponential functions have an asymptote, or a line that the graph gets closer and closer to, but never crosses or touches. Every exponential function has a y-intercept, where the graph crosses the y-axis. There is, at most, one x-intercept, where the function crosses the x-axis. 3.4.2: Graphing Exponential Functions

  2. Key Concepts To find the y-intercept of an exponential function, evaluate f(0). The y-intercept has the coordinates (0, f(0)). To locate the y-intercept of a graphed function, determine the coordinates of the function where the line crosses the y-axis. To find the x-intercept in function notation, set f(x) = 0 and solve for x. The x-intercept has the coordinates (x, 0). 3.4.2: Graphing Exponential Functions

  3. Key Concepts, continued To locate the x-intercept of a graphed function, determine the coordinates of the line where the line crosses the x-axis. Not all exponential functions cross the x-axis. The asymptote of exponential functions of the form f(x) = abx is always the x-axis, or y = 0. If the exponential function is of the form f(x) = abx + k, then the function will be shifted vertically by the same number of units as k. 3.4.2: Graphing Exponential Functions

  4. Key Concepts, continued The asymptote is then y = k. The end behavior, or the behavior of the graph as xbecomes larger or smaller, will always be one of three descriptions: infinity, negative infinity, or the asymptote. It is easiest to first graph the function and then observe what happens to the value of y as the value of x increases and decreases. Graph complex exponential models using technology as values can become quite large or small very quickly. 3.4.2: Graphing Exponential Functions

  5. Key Concepts, continued Graphing Equations Using a TI-83/84: Step 1: Press [Y=]. Step 2: Key in theequationusing [X, T, q, n] forx. Step 3: Press [WINDOW] tochangetheviewingwindow, ifnecessary. Step 4: Enter in appropriatevaluesfor Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl, usingthearrowkeystonavigate. Step 5: Press [GRAPH]. 3.4.2: Graphing Exponential Functions

  6. Key Concepts, continued Graphing Equations Using a TI-Nspire: Step 1: Press the home key. Step 2: Arrow over to the graphing icon (the picture of the parabola or the U-shaped curve) and press [enter]. Step 3: Enter in the equation and press [enter]. Step 4: To change the viewing window: press [menu], arrow down to number 4: Window/Zoom, and click the center button of the navigation pad. 3.4.2: Graphing Exponential Functions

  7. Key Concepts, continued Step 5: Choose 1: Window settings by pressing the center button. Step 6: Enter in the appropriate XMin, XMax, YMin, and YMax fields. Step 7: Leave the XScale and YScale set to auto. Step 8: Use [tab] to navigate among the fields. Step 9: Press [tab] to “OK” when done and press [enter]. 3.4.2: Graphing Exponential Functions

  8. Common Errors/Misconceptions incorrectly plotting points mistaking the y-intercept for the x-intercept and vice versa not being able to identify key features of an exponential model confusing the value of a function for its corresponding x-coordinate 3.4.2: Graphing Exponential Functions

  9. Guided Practice Example 2 Create a table of values for the exponential function f(x) = –1(3)x – 2. Identify the asymptote and y-intercept of the function. Plot the points and sketch the graph of the function, and describe the end behavior. 3.4.2: Graphing Exponential Functions

  10. Guided Practice: Example 2, continued Create a table of values. Choose values of x and solve for the corresponding values of f(x). 3.4.2: Graphing Exponential Functions

  11. Guided Practice: Example 2, continued Identify the asymptote of the function. The asymptote of the function is always the constant, k. In the function f(x) = –1(3)x – 2, the value of k is –2. The asymptote of the function is y = –2. 3.4.2: Graphing Exponential Functions

  12. Guided Practice: Example 2, continued Determine the y-intercept of the function. The y-intercept of the function is the value of f(x) when x is equal to 0. It can be seen in the table that when x = 0, f(x) = –3. The y-intercept is (0, –3). 3.4.2: Graphing Exponential Functions

  13. Guided Practice: Example 2, continued Graph the function. Use the table of values to create a graph of the function. 3.4.2: Graphing Exponential Functions

  14. Guided Practice: Example 2, continued Describe the end behavior of the graph. The end behavior is what happens at the ends of the graph. As x becomes larger, the value of the function approaches negative infinity. As x becomes smaller, the value of the function approaches the asymptote, –2. ✔ 3.4.2: Graphing Exponential Functions

  15. Guided Practice: Example 2, continued 15 3.4.2: Graphing Exponential Functions

  16. Guided Practice Example 3 Create a table of values for the exponential function . Identify the asymptote and y-intercept of the function. Plot the points and sketch the graph of the function, and describe the end behavior. 3.4.2: Graphing Exponential Functions

  17. Guided Practice: Example 3, continued Create a table of values. Choose values of x and solve for the corresponding values of f(x). 3.4.2: Graphing Exponential Functions

  18. Guided Practice: Example 3, continued Identify the asymptote of the function. The asymptote of the function is always the constant, k. In the function , the value of k is –3. The asymptote of the function is y = –3. 3.4.2: Graphing Exponential Functions

  19. Guided Practice: Example 3, continued Determine the y-intercept of the function. The y-intercept of the function is the value of f(x) when x is equal to 0. It can be seen in the table that when x = 0, f(x) = 1. The y-intercept is (0, 1). 3.4.2: Graphing Exponential Functions

  20. Guided Practice: Example 3, continued Graph the function. Use the table of values to create a graph of the function. 3.4.2: Graphing Exponential Functions

  21. Guided Practice: Example 3, continued Describe the end behavior of the graph. The end behavior is what happens at the ends of the graph. As x becomes larger, the value of the function approaches the asymptote, –3. As x becomes smaller, the value of the function approaches infinity. Since the function approaches infinity as x becomes smaller, the graph shows exponential decay. ✔ 3.4.2: Graphing Exponential Functions

  22. Guided Practice: Example 3, continued 3.4.2: Graphing Exponential Functions

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