Exponential functions have the form $latex y={{b}^x}$, where $latex b> 0$. Exponential functions are functions that remain proportional to their original value as it increases or decreases. If b>1, the function increases and if 1>b>0, the function decreases.

In this article, we will learn about the graphs of exponential functions and learn about their properties. We will also look at some practice examples.

## Definition of exponential functions

In its most basic form, an exponential function can be thought of as a function where the variable appears in the exponent. The simplest exponential function is a function of the form $latex y={{b}^x}$, where *b* is a positive number.

When we have $latex b>1$, the function grows in a way that is proportional to its original value. This is called exponential growth.

When we have $latex 1>b>0$, the function decreases in a way that is proportional to its original value. This is called exponential decay.

## Graphs of exponential functions

Let’s look at the following examples of how to graph exponential functions.

### EXAMPLE 1

Suppose we want to graph the function $latex y={{2} ^x}$. One way to accomplish this is to choose specific values for *x* and use them in the function to generate values for *y*. By doing so, we can obtain the following points:

$latex \left( {-2,\frac{1}{4}} \right),~\left( {-1,\frac{1}{2}} \right),~\left( {0,~1} \right),~\left( {1,~2} \right)$ y $latex ( {2,~4}).$

As we connect these points, we can see that we form a curve that crosses the *y*-axis through the point (0, 1). The graph grows as the values of *x* get larger. We can see that this graph tends to infinity as the values of *x* go to infinity.

Similarly, we can see that as the values of *x* get smaller and smaller, the curve gets closer and closer to the *x*-axis. The curve approaches zero as the values of *x* approach negative infinity. Since the curve never touches the *x*-axis, we have that the *x*-axis is a horizontal asymptote of the function.

For any exponential function of the form $latex y={{b}^x}$, where we have $latex b> 1$, we have the point (1, b) on the graph.

### EXAMPLE 2

Now, consider the function $latex y={{(\frac{1}{2})}^x}$ when $latex 1> b> 0$. Similar to the previous example, we can graph this exponential function by using several values of *x* and substituting them in the function to obtain values of *y.* We can form the following Cartesian coordinates:

$latex \left( {-2, 4} \right),~\left( {-1, 2} \right),~\left( {0,~1} \right),~\left( {1,~\frac{1}{2}} \right)$ y $latex ( {2, \frac{1}{4}}).$

When we connect the points represented by those Cartesian coordinates, we see that the curve that is formed crosses the *y*-axis through the point (0, 1). The graph of this exponential function decreases as the values of *x* gets larger.

The graph approaches zero as the values of *x* tend to infinity. This means that the graph has a horizontal asymptote on the *x*-axis. We can also observe that the graph tends to infinity as the values of *x* tend to negative infinity.

For any exponential function of the form $latex y={{b}^x}$, where we have $latex 1>b>0$, we have the point (1, *b*) on the graph.

By comparing their graphs, we can see that the graph of $latex y=(\frac{1}{2})^x$ is symmetric to the graph of $latex y={{2}^x}$ with respect to the *y* axis:

## Limitation of *b* to positive numbers

The value of *b* is limited to numbers greater than 1 due to the following reasons:

**•** If we have $latex b=1$, the function becomes $latex y={{1}^x}$. We know that 1 raised to any power equals 1, so we would actually have the function $latex y=1$. This function produces a horizontal line, so we would not obtain an exponential function.

**• **If *b* is negative, then when we raise *b* to an even power, we will get positive numbers. However, when we raise *b* to an odd power, we will get negative numbers. This means that it is impossible to connect the points in a meaningful way, so we will not get a shape similar to the graphs shown above.

## Properties of graphs of exponential functions

The following are some of the properties that all exponential graphs share:

**•** The point (0, 1) is always on the graph of the exponential function of the form $latex y={{b}^x}$, because *b* is a positive number and all positive numbers raised to the power of zero are equal to 1.

**•** The point (1, *b*) is always on the graph of the exponential function of the form $latex y={{b}^x}$. This is because any number *b* raised to the power of 1 equals *b*.

**•** The function $latex y={{b}^x}$ will always produce positive values. Since *b* is a positive number, we will always get positive values when raised to any power.

**•** The *x*-axis is a horizontal asymptote of the function $latex y={{b}^x}$ because the function will always approach the *x*-axis as *x* approaches positive or negative infinity, but it never crosses the *x*-axis since the function is never equal to 0.

## See also

Interested in learning more about graphs of functions? Take a look at these pages:

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