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x y f(x) = As the base increases above 1, the function turns more quickly to the right as it rises. As the base decreases from 1 to 0, the function turns more quickly to the right as it falls. The closer the base is to 1, the closer the function is to a vertically straight line through x = 1. For each of these functions: The asymptote is x = 0. Domain = ]0, ∞ Range = For functions f(x) and h(x), functions are increasing. For functions g(x) and k(x), functions are decreasing. log c x

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x y x y x = 2 y 1 0 x = 2 0 2 1 x = 2 1 4 2 x = 2 2 8 3 x = 2 3 0.5 x = 2 -1 x y x = 2 y - 2 0 x = 2 0 - 20 1 x = 2 1 - 22 2 x = 2 2 - 26 3 x = 2 3 - 2-1.5 x = 2 -1 - 2 When h = -2, the function shifts 2 to the left. When h = 3, the function shifts 3 to the right. x = -2 x = 0 x = 3

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The logarithmic function can have up to 5 different parameters: a, b, c, h, k. f(x) = a log c (b(x - h)) + k Each of these parameters impact the appearance of the graph. h and k are the shifting parameters, whereas a and b are scale change parameters and parameter c can affect the shape of the curve like parameter a. By studying the impact of these parameters, we can anticipate some of its characteristics and develop rules to determine some of its features. We have already seen the impact that parameter c has on the graph. Now it is time to look at the other parameters.

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x y x = 2 y 1 0 x = 2 0 2 1 x = 2 1 4 2 x = 2 2 8 3 x = 2 3 0.5 x = 2 -1 x y x y x = 2 y-2 When k = 2, the function shifts 2 up. 1 2 x = 2 2-2 2 3 x = 2 3-2 4 4 x = 2 4-2 8 5 x = 2 5-2 0.5 1 x = 2 1-2 When k = -3, the function shifts 3 down. The asymptote doesn’t change. It remains x = 0. x = 0

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x y x = 2 y 1 0 x = 2 0 2 1 x = 2 1 4 2 x = 2 2 8 3 x = 2 3 0.5 x = 2 -1 x y x -0.5 -2 -3 1 y 0 1 2 3 -1 x = 0 y 0 1 2 3 -1 x 0.5 1 2 4 0.25 As b increases the function turns LESS sharply. If b < 0, the graph appears to the left of the asymptote.

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x y x = 2 y 1 0 x = 2 0 2 1 x = 2 1 4 2 x = 2 2 8 3 x = 2 3 0.5 x = 2 -1 y 0 2 4 6 -2 x y y 0 2 -2 -4 -6 x 1 0.5 2 4 8 x = 0 x 1 2 4 8 0.5 As a increases the function turns LESS sharply. If a < 0, the graph becomes decreasing graph even though c > 0.

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Parameter c can affect whether the function will increase or decrease – but not by itself. Parameters a and b also affect the graph in that way. 0 < c < 1 c > 1 ab > 0ab < 0 INCREASINGDECREASING INCREASINGDECREASING Parameter h causes the function to shift horizontally. k > 0 k < 0Shifts Down Shifts Uph > 0Shifts Right h < 0Shifts Left Parameter k causes the function to shift vertically. Asymptote shifts too. The logarithmic function can have up to 5 different parameters: a, b, c, h, k. f(x) = a log c (b(x - h) ) + k

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1 Example 3 (a) Find the range of f(x) = 1/x with domain [1, ). Solution The function f is decreasing on the interval [1, ) from its largest value.

1 Example 3 (a) Find the range of f(x) = 1/x with domain [1, ). Solution The function f is decreasing on the interval [1, ) from its largest value.

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