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Introduction An exponential function has various key features, or characteristics. Two of these are the domain, which is the set of all input values (x-values) that satisfy the given function without restriction, and the the set of all outputs (y-values) that are valid for the function. Other key features include the y-intercept(s), asymptote(s), and end behavior. The y-intercept is the point at which the graph of a function crosses the y-axis, written (0, y). The asymptote is an equation that represents a set of points that are not allowed by the conditions in a parent function or model; shown on a graph, the asymptote is a line that a function gets closer and closer to, but never touches. 1 4.1.2: Properties of Exponential Functions Introduction, continued The end behavior is the behavior of the graph as x approaches positive or negative infinity, and can be described as whether the function’s graph increases or decreases within its domain. These key features can be determined by analyzing the values of a, b, and c in the general form of the equation of the function, or by viewing a graph of the function. 2 4.1.2: Properties of Exponential Functions Key Concepts • Recall that the general form of an exponential function is f(x) = a(bx) + c, where a, b, and c are constants and b is greater than 0 but not equal to 1. • If a = 1 and c = 0, f(x) = a(bx) + c can be rewritten as f(x) = bx. • The exponential function f(x) = bx has a domain of all real numbers, a range of all real numbers greater than 0, a y-intercept of 1, and an asymptote that is the x-axis. 3 4.1.2: Properties of Exponential Functions Key Concepts, continued • In general, if a is positive, a function of the form f(x) = a(bx) + c has a domain of all real numbers, a range of all real numbers greater than c, a y-intercept of a + c, and an asymptote of y = c. • If b > 1, the function increases within its domain, and if 0 < b < 1, the function decreases within its domain. In other words, the function shows exponential growth if b > 1 and exponential decay if 0 < b < 1. • The domain and range of an exponential function may be restricted if the function is used to model a realworld scenario. 4.1.2: Properties of Exponential Functions 4 Key Concepts, continued • The graph of an exponential function can be obtained by using a graphing calculator. On a TI-83/84: Step 1: Press [MODE]. Step 2: Make sure Func is highlighted on the fourth line. If not, arrow down to it and press [ENTER]. Step 3: Press [Y=]. Press [CLEAR] to delete any equations. Step 4: Enter the equation of the exponential function at Y1. Use [X, T, θ, n] for variables and [^] for exponents. (continued) 4.1.2: Properties of Exponential Functions 5 Key Concepts, continued Step 5: If needed, press [WINDOW] to adjust the viewing window. Change the settings as appropriate. Step 6: Press [GRAPH]. On a TI-Nspire: Step 1: Press [home]. Arrow over to the graphing icon and press [enter]. Step 2: Press [menu]. Use the arrow key to select 3: Graph Type and then 1: Function. Press [enter]. Step 3: Use your keypad to enter the equation of the exponential function after f1(x). (continued) 4.1.2: Properties of Exponential Functions 6 Key Concepts, continued Step 4: If needed, adjust the viewing window. Press [menu], then select 4: Window/Zoom, and then 1: Window Settings. Change the settings as appropriate. Step 5: Press [enter]. 7 4.1.2: Properties of Exponential Functions Common Errors/Misconceptions • not taking into account the values of a and c in an exponential function of the form f(x) = a(bx) + c when determining the function’s range, y-intercept, and/or asymptote • not restricting the domain and range of an exponential function given a real-world scenario • forgetting to switch a graphing calculator to function mode • setting a graphing calculator’s window variables inappropriately 8 4.1.2: Properties of Exponential Functions Guided Practice Example 1 What are the domain, range, y-intercept, asymptote, and end behavior of the exponential function f(x) = 7x? Confirm your results by graphing. 9 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued 1. Determine the values of a, b, and c in the function. The function is in the form f(x) = a(bx) + c, so a = 1, b = 7, and c = 0. 10 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued 2. Use the values of a and c to find the domain, range, y-intercept, and asymptote of the function. From the previous step, it is known that a = 1 and c = 0. Recall that if a is positive, a function of the form f(x) = a(bx) + c has a domain of all real numbers, a range of all real numbers greater than c, a y-intercept of a + c, and an asymptote of y = c. Since a is positive, the domain is all real numbers. The range is all real numbers greater than c, which is 0. 11 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued The y-intercept is a + c, or 1 + 0, which simplifies to 1. The asymptote is y = c, or y = 0. Therefore, the domain of f(x) = 7x is all real numbers, the range is all real numbers greater than 0, the y-intercept is 1, and the asymptote is the x-axis (y = 0). 12 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued 3. Use the value of b to determine the function’s end behavior. The value of b in the function f(x) = 7x is 7, and 7 > 1. When any number greater than 1 is raised to a larger and larger power, the resulting value also becomes larger and larger. Therefore, the function increases within its domain. 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued 4. Graph the function and use it to confirm your findings. Graph the function f(x) = 7x using the directions appropriate to your calculator model. Compare the key features of the graph (domain, range, y-intercept, asymptote, and end behavior) with your findings. On a TI-83/84: Step 1: Press [MODE]. Step 2: Make sure Func is highlighted on the fourth line. If not, arrow down to it and press [ENTER]. (continued) 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued Step 3: Press [Y=]. Press [CLEAR] to delete any equations. Step 4: Enter the equation of the exponential function at Y1. Use [X, T, θ, n] for variables and [^] for exponents. Step 5: If needed, press [WINDOW] to adjust the viewing window. Change the settings as appropriate. Step 6: Press [GRAPH]. (continued) 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued On a TI-Nspire: Step 1: Press [home]. Arrow over to the graphing icon and press [enter]. Step 2: Press [menu]. Use the arrow key to select 3: Graph Type and then 1: Function. Press [enter]. Step 3: Use your keypad to enter the equation of the exponential function after f1(x). Step 4: If needed, adjust the viewing window. Press [menu], then select 4: Window/Zoom, and then 1: Window Settings. Change the settings as appropriate. Step 5: Press [enter]. 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued The graph extends infinitely in both directions, so the domain is all real numbers. The graph includes only y-values greater than 0, which means the range is all real numbers greater than 0. 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued The graph intersects the y-axis at (0, 1), so the y-intercept is 1. The graph approaches but never touches the x-axis; therefore, the asymptote is the x-axis (y = 0). This graph shows the function values rising from left to right, so the function is increasing within its domain. Thus, the graph confirms the findings from the previous steps. ✔ 4.1.2: Properties of Exponential Functions Guided Practice: Example 1, continued 19 4.1.2: Properties of Exponential Functions Guided Practice Example 2 What are the domain, range, y-intercept, asymptote, and x end behavior of the exponential function g(x) = 6 æ1æ ? + 5 æ æ8 æ æ Confirm your results by graphing. 20 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued 1. Determine the values of a, b, and c in the function. The function is in the form f(x) = a(bx) + c, so a = 6, 1 b = , and c = 5. 8 21 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued 2. Use the values of a and c to find the domain, range, y-intercept, and asymptote of the function. From the previous step, it is known that a = 6 and c = 5. Recall that if a is positive, a function of the form f(x) = a(bx) + c has a domain of all real numbers, a range of all real numbers greater than c, a y-intercept of a + c, and an asymptote of y = c. 22 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued Since a is positive, the function follows this rule. Therefore, the domain is all real numbers, the range is all real numbers greater than 5, the y-intercept is 6 + 5 = 11, and the asymptote is y = 5. 23 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued 3. Use the value of b to determine the function’s end behavior. The value of b is 1 , and 0 < 1 < 1 . When any number 8 8 greater than 0 and less than 1 is raised to a larger and larger power, the resulting values become smaller and smaller. Therefore, the function decreases within its domain. 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued 4. Graph the function and use it to confirm your findings. x æ1æ Graph the function g(x) = 6 æ æ + 5 on a graphing æ8 æ calculator. 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued The graph confirms that the domain is all real numbers, the range is all real numbers greater than 5, the y-intercept is 11, and the asymptote is y = 5. ✔ 4.1.2: Properties of Exponential Functions Guided Practice: Example 2, continued 27 4.1.2: Properties of Exponential Functions