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Exponential Functions

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Exponential Functions

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Exponential Functions

Exponential Functions

Exponential Functions

Exponential Functions

Exponential Functions

Exponents can be variables. Variable exponents obey all the properties of exponents listed in Properties of Exponents.

An exponential function is a function that contains a variable exponent. For example, f (x) = 2x and g(x) = 5ƒ3x are exponential functions. We can graph exponential functions. Here is the graph of f (x) = 2x :

Figure %: f (x) = 2x
The graph has a horizontal asymptote at y = 0 , because 2x > 0 for all x . It passes through the point (0, 1) .

We can translate this graph. For example, we can shift the graph down 3 units and left 5 units. Here is the graph of f (x) = 2x+5 - 3 :

Figure %: f (x) = 2x+5 - 3
This graph has a horizontal asymptote at y = - 3 and passes through the point (- 5, - 2) .

We can stretch and shrink the graph vertically by multiplying the output by a constant--see Stretches. For example, f (x) = 3ƒ2x is stretched vertically by a factor of 3:

Figure %: f (x) = 3ƒ2x
This graph has a horizontal asymptote at y = 0 and passes through the point (0, 3) .

We can also graph exponential functions with other bases, such as f (x) = 3x and f (x) = 4x . We can think of these graphs as differing from the graph of f (x) = 2x by a horizontal stretch or shrink: when we multiply the input of f (x) = 2x by 2, we get f (x) = 22x = (22)x = 4x . Thus, the graph of f (x) = 4x is shrunk horizontally by a factor of 2 from f (x) = 2x :

Figure %: f (x) = 4x
This graph has a horizontal asymptote at y = 0 and passes through the point (0, 1) .

The graph of f (x) = a x does not always differ from f (x) = 2x by a rational factor. Thus, it is useful to think of each base individually, and to think of a different base as a horizontal stretch for comparison purposes only.

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