how to find asymptotes of exponential function

Exponential Functions

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

An exponential function is a office that contains a variable exponent. For example, f (x) = ii^x and g(x) = 5ƒ3^x are exponential functions. We tin graph exponential functions. Hither is the graph of f (x) = 2^x :

The graph has a horizontal asymptote at y = 0, because ii¹⁰ > 0 for all x . It passes through the point (0, 1).

We can translate this graph. For example, we tin can shift the graph downwardly iii units and left 5 units. Hither is the graph of f (ten) = ii^x+5 - 3:

This graph has a horizontal asymptote at y = - 3 and passes through the signal (- 5, - 2).

We tin can stretch and shrink the graph vertically by multiplying the output by a constant--see Stretches. For instance, f (x) = 3ƒ2^x is stretched vertically by a cistron of 3:

This graph has a horizontal asymptote at y = 0 and passes through the point (0, 3).

We can too graph exponential functions with other bases, such as f (x) = 3¹⁰ and f (x) = iv^x . We can think of these graphs every bit differing from the graph of f (ten) = 2¹⁰ past a horizontal stretch or shrink: when we multiply the input of f (x) = ii^x by 2, nosotros become f (x) = ii^2x = (2²)^x = iv^x . Thus, the graph of f (x) = 4^x is shrunk horizontally by a factor of 2 from f (x) = ii^x :

This graph has a horizontal asymptote at y = 0 and passes through the signal (0, 1).

The graph of f (x) = a ^x does not ever differ from f (x) = 2^x by a rational factor. Thus, it is useful to retrieve of each base of operations individually, and to recall of a different base equally a horizontal stretch for comparison purposes only.

The graph of an exponential function can also be reflected over the 10 -axis or the y -axis, and rotated around the origin, as in Heading .

The full general form of an exponential function is f (10) = cƒa ^x-h + k , where a is a positive constant and a≠i. a is called the base. The graph has a horizontal asymptote of y = k and passes through the point (h, c + k).