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8.1 Exponential Growth Goal: Graph exponential growth functions.

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Warm-up Evaluate without using a calculator:

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Exponential Growth An exponential function has the form, where a 0 and the base b is a positive number other than 1. If a > 0 and b > 1, such a function is an exponential growth function, because the values of y grow as x increases. Graph the function. Make a table: XY -2 0 1 2 3

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Exponential Growth Graphs Characteristics of the graph of y = ab x when a > 0 and b > 1 The graph rises from left to right. The graph passes through (0, a) and (1, ab) The domain is all real numbers. The range is y > 0 An asymptote is a line that a graph approaches as x goes to positive or negative infinity. The x-axis is an asymptote of the graph of y = ab x. (0, a) (1, ab)

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Example Graph y = 5 x y = (1)5 x so a = 1, and b = 5 Therefore the graph will go through (0, 1) and (1, 5) The x-axis is an asymptote so the graph will not intersect it.

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Example Graph y = ()3 x a = , and b =3 Therefore the graph will go through (0, ) and (1, 1) The x-axis is an asymptote so the graph will not intersect it. Other points that it passes through are (2, 3) and (3, 9).

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Example Graph y = (2)3 x+1 - 2 This has the form y = ab x - h + k a = 2, b =3, h = -1, k = -2. Think of moving the y-axis 1 unit left, and the x-axis 2 units down. The graph will go through the points (h, a + k) and (h+1, ab+k). So the graph goes through (-1, 0) and (0, 4). The line y = k is an asymptote, so the graph gets closer to the line y =-2 without intersecting it.

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Example Graph y = (1/2)4 x +2 This has the form y = ab x - h + k a = 1/2, b =4, h = 0, k = 2. Think of moving the x-axis 2 units up. The graph will go through the points (h, a + k) and (h+1, ab+k). So the graph goes through (0, 2-1/2) and (1, 4). The line y = k is an asymptote, so the graph gets closer to the line y =2 without intersecting it.

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Assignment Worksheet 8.1