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If a and b are positive real numbers and b is not equal to 1, how does the graph of f(x) = ab^x change when b is changed?

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The graph of the function f(x) = ab^x depends on the values of a and b.

When a is held constant, changing b will cause the graph to either stretch or compress horizontally, depending on whether b is greater than or less than 1.

If b is greater than 1, the function will grow faster as x increases, causing the graph to stretch horizontally. The larger the value of b, the faster the function will grow. For example, consider the following graphs of the function f(x) = 2(1.5)^x and f(x) = 2(2)^x:

Graph of f(x) = 2(1.5)^x and f(x) = 2(2)^x

As we can see, the graph of f(x) = 2(2)^x grows faster than the graph of f(x) = 2(1.5)^x, causing it to stretch more horizontally.

On the other hand, if b is less than 1, the function will grow slower as x increases, causing the graph to compress horizontally. The smaller the value of b, the slower the function will grow. For example, consider the following graphs of the function f(x) = 2(0.5)^x and f(x) = 2(0.2)^x:

Graph of f(x) = 2(0.5)^x and f(x) = 2(0.2)^x

As we can see, the graph of f(x) = 2(0.2)^x compresses more horizontally than the graph of f(x) = 2(0.5)^x.

In summary, changing the value of b in the function f(x) = ab^x will cause the graph to stretch or compress horizontally, depending on whether b is greater than or less than 1. If b is greater than 1, the graph will stretch horizontally and if b is less than 1, the graph will compress horizontally.
User Hugh Fisher
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