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Introduction Exponential functions are functions that can be written in the form f(x) = ab x, where a is the initial value, b is the rate of decay or growth, x is the time, and f(x) is the final output value. The growth factor is the multiple by which a quantity increases or decreases over time. The rate of change of an exponential function can be calculated using the formula, over a specified interval. An interval is a continuous series of values. 1 3.5.2: Comparing Exponential Functions

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Introduction, continued The y-intercept is the point at which the function crosses the y-axis and has the point (0, y). Both the rate of change and y-intercept can be determined from tables, equations, and graphs. Exponential functions can also be compared to one another using these features. 2 3.5.2: Comparing Exponential Functions

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Key Concepts Exponential functions can be represented in words or as equations, graphs, or tables. To compare exponential functions, determine the rate of change and the intercepts of each function. Review the following processes for identifying the rate of change and the y-intercept of an exponential function. 3 3.5.2: Comparing Exponential Functions

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Key Concepts, continued 4 3.5.2: Comparing Exponential Functions Identifying the Rate of Change and the y-intercept from Context 1.Determine the interval to be observed. 2.Create a table of values by choosing appropriate x-values, substituting them, and solving for f(x). 3.Choose two points from the table. 4.Assign one point to be (x 1, y 1 ) and the other point to be (x 2, y 2 ). 5.Substitute the values into the slope formula,. 6.The result is the rate of change for the interval between the two points chosen. 7.Determine which information tells you the y-intercept, or b. This could be an initial value or a starting value, a flat fee, and so forth.

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Key Concepts, continued 5 3.5.2: Comparing Exponential Functions Identifying the Rate of Change and the y-intercept from Exponential Equations 1.Determine the interval to be observed. 2.Determine (x 1, y 1 ) by identifying the starting x-value of the interval and substituting it into the function. 3.Solve for f(x). 4.Determine (x 2, y 2 ) by identifying the ending x-value of the interval and substituting it into the function. 5.Solve for f(x). 6.Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula,, to calculate the rate of change. 7.Determine the y-intercept by substituting 0 for x and solving for f(x).

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Key Concepts, continued 6 3.5.2: Comparing Exponential Functions Identifying the Rate of Change and the y-intercept from a Table 1.Determine the interval to be observed. 2.Assign one point to be (x 1, y 1 ) and the other point to be (x 2, y 2 ). 3.Substitute the values into the slope formula,. 4.The result is the rate of change for the interval between the two points chosen. 5.Identify the y-intercept as the coordinate in the form (0, y).

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Key Concepts, continued 7 3.5.2: Comparing Exponential Functions Identifying the Rate of Change and the y-intercept from a Graph 1.Determine the interval to be observed. 2.Identify (x 1, y 1 ) as the starting point of the interval. 3.Identify (x 2, y 2 ) as the ending point of the interval. 4.Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula,, to calculate the rate of change. 5.Identify the y-intercept as the coordinate in the form (0, y).

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Key Concepts, continued You can compare the functions once you have identified the rate of change and the y-intercept of each function. Exponential functions are increasing if the rate of change is a positive value. Exponential functions are decreasing if the rate of change is a negative value. The greater the rate of change, the steeper the line connecting the points of the interval will appear on the graph. 8 3.5.2: Comparing Exponential Functions

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Common Errors/Misconceptions incorrectly determining the rate of change assuming that a positive slope will be steeper than a negative slope comparing functions over different intervals incorrectly applying the order of operations using the exponential growth model instead of exponential decay and vice versa 9 3.5.2: Comparing Exponential Functions

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Guided Practice Example 1 Compare the properties of each of the two functions on the next slide over the interval [0, 16]. 10 3.5.2: Comparing Exponential Functions

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Guided Practice: Example 1, continued 11 3.5.2: Comparing Exponential Functions xg(x)g(x) 0850 4976.55 81121.94 121288.98 161480.88 Function A Function B

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Guided Practice: Example 1, continued 1.Compare the y-intercepts of each function. Identify the y-intercept of the graphed function, f(x). The graphed function appears to cross the y-axis at the point (0, 850). According to the table, g(x) has a y-intercept of (0, 850). Both functions have a y-intercept of (0, 850). 12 3.5.2: Comparing Exponential Functions

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Guided Practice: Example 1, continued 2.Compare the rate of change for each function over the interval [0, 16]. Calculate the rate of change over the interval [0, 16] for f(x). Let (x 1, y 1 ) = (0, 850). Determine (x 2, y 2 ) from the graph. The value of y when x is 16 is approximately 1,600. Let (x 2, y 2 ) = (16, 1600). Calculate the rate of change using the slope formula. 13 3.5.2: Comparing Exponential Functions

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14 3.5.2: Comparing Exponential Functions Guided Practice: Example 1, continued Slope formula Substitute (0, 850) and (16, 1600) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. The rate of change for f(x) is approximately 47.

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15 3.5.2: Comparing Exponential Functions Guided Practice: Example 1, continued Calculate the rate of change over the interval [0, 16] for g(x). Let (x 1, y 1 ) = (0, 850). Determine (x 2, y 2 ) from the table. The value of y when x is 16 is 1,480.88. Let (x 2, y 2 ) = (16, 1480.88). Calculate the rate of change using the slope formula.

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16 3.5.2: Comparing Exponential Functions Guided Practice: Example 1, continued Slope formula Substitute (0, 850) and (16, 1480.88) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. The rate of change for g(x) is 39.43.

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17 3.5.2: Comparing Exponential Functions Guided Practice: Example 1, continued The rate of change for the graphed function, f(x), is greater over the interval [0, 16] than the rate of change for the function in the table, g(x).

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Guided Practice: Example 1, continued 3.Summarize your findings. The y-intercepts of both functions are the same; however, the graphed function, f(x), has a greater rate of change over the interval [0, 16]. 18 3.5.2: Comparing Exponential Functions ✔

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19 3.5.2: Comparing Exponential Functions Guided Practice: Example 1, continued 19

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Guided Practice Example 3 A pendulum swings to 90% of its previous height. Pendulum A starts at a height of 50 centimeters. Its height at each swing is modeled by the function f(x) = 50(0.90) x. The height after every fifth swing of Pendulum B is recorded in the following table. Compare the properties of each function over the interval [5, 15]. 20 3.5.2: Comparing Exponential Functions

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Guided Practice: Example 3, continued 21 3.5.2: Comparing Exponential Functions xf(x)f(x) 0100 559.05 1034.87 1520.59 2012.16

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Guided Practice: Example 3, continued 1.Compare the y-intercepts of each function. Identify the y-intercept of Pendulum A. The problem states that the pendulum starts at a height of 50 centimeters. The y-intercept of the function is (0, 50). Identify the y-intercept of Pendulum B. The value of f(x) is 100 when x is 0. The y-intercept of the function is (0, 100). 22 3.5.2: Comparing Exponential Functions

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Guided Practice: Example 3, continued 2.Compare the rate of change for each function over the interval [5, 15]. Calculate the rate of change over the interval [5, 15] for Pendulum A. Determine (x 1, y 1 ) from the function. f(x) = 50(0.90) x Original function f(5) = 50(0.90) 5 Substitute 5 for x. f(5) = 29.52 Simplify as needed. Let (x 1, y 1 ) = (5, 29.52). 23 3.5.2: Comparing Exponential Functions

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Guided Practice: Example 3, continued Determine (x 2, y 2 ) from the function. f(x) = 50(0.90) x Original function f(15) = 50(0.90) 15 Substitute 15 for x. f(15) ≈ 10.29 Simplify as needed. The value of y when x is 15 is approximately 10.29. Let (x 2, y 2 ) = (15, 10.29). 24 3.5.2: Comparing Exponential Functions

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25 3.5.2: Comparing Exponential Functions Guided Practice: Example 3, continued Calculate the rate of change using the slope formula. Slope formula Substitute (5, 29.52) and (15, 10.29) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. The rate of change for Pendulum A’s function is approximately –1.923 centimeters per swing.

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26 3.5.2: Comparing Exponential Functions Guided Practice: Example 3, continued Calculate the rate of change over the interval [5, 15] for Pendulum B. Let (x 1, y 1 ) = (5, 59.05). Let (x 2, y 2 ) = (15, 20.59). Calculate the rate of change using the slope formula.

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27 3.5.2: Comparing Exponential Functions Guided Practice: Example 3, continued Slope formula Substitute (5, 59.05) and (15, 20.59) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed.

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28 3.5.2: Comparing Exponential Functions Guided Practice: Example 3, continued The rate of change for Pendulum B’s function is approximately –3.846 centimeters per swing. The rate of change for Pendulum B is greater over the interval [5, 15] than the rate of change for Pendulum A.

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Guided Practice: Example 3, continued 3.Summarize your findings. The y-intercept of Pendulum A is less than the y-intercept of Pendulum B. This means that Pendulum B begins higher than Pendulum A. The rate of change for Pendulum A is less than the rate of change for Pendulum B. This means that Pendulum B is losing height faster than Pendulum A. 29 3.5.2: Comparing Exponential Functions ✔

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30 3.5.2: Comparing Exponential Functions Guided Practice: Example 3, continued

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