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# Recognizing Exponential Functions

Exponential functions are another type of function family. Similar to linear functions, it is their rates of change that make them distinct. Additionally, the appearance of the corresponding function rules and graphs are unique to this family.
Rule

## Rate of Change of an Exponential Function

For linear functions, the rate of change is constant, meaning that for each step in the $x$-direction, the change between $y$-values is the same. The rate of change for exponential functions, however, is not constant. Consider the following tables with the same $x$-values.

The difference between the $y$-values in the left table is constant ($+2$), but that's not the case for the right table.

Therefore, the left table represents a linear function, whereas the right table represents a non-linear function. Notice how, for the table on the right, the $y$-values double for each step in the $x$-direction. \begin{aligned} &1\cdot 2 = 2 \quad 2\cdot2 = 4 \quad 4\cdot2=8 \\[0.5em] & \quad \quad 8\cdot 2 = 16 \quad 16\cdot 2 = 32 \end{aligned} Thus, every $y$-value is multiplied by $2$ to get to the next.

While the left table corresponds to a function with a constant increase, the right table corresponds to a function with a constant multiplier. This type of function is an exponential function.
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Exercise

The tables represent three different functions. Determine which is exponential.

Show Solution
Solution

An exponential function changes by a constant multiplier with each step in the $x$-direction. Notice that all tables have the same $x$-values, which increase by $1.$

Next, we can analyze the $y$-values for each table individually. Let's start with the second table. Notice that the $y$-values increase by $3$ each step. This means that the rate of change is constant. Therefore, the function is linear, and thus, not exponential.

Let's move to the first table. It can be seen that the $y$-values do not increase with a constant rate of change because $6 + {\color{#0000FF}{12}} = 18 \ \text{and} \ 18 + {\color{#009600}{27}} = 54.$ Since we know this rate of change is not constant, we can now try to determine if it changes by a constant multiplier. To determine what number multiplies down the table, we can divide up the table. $\frac{486}{162}=3 \quad \frac{162}{54}=3 \quad \frac{54}{18}=3 \quad \frac{18}{6}=3$ It appears that the constant multiplier is $3.$ Thus, the first table corresponds to an exponential function.

Looking at the third table, it is apparent that the function does not have a constant rate of change. We can use the same method as above to determine if the $y$-values change by a constant multiplier. $\frac{125}{64}=1.95 \quad \frac{64}{27}=2.37 \quad \frac{27}{8}=3.38 \quad \frac{8}{1}=8$ The third table does not change by a constant multiplier. Thus, it is not an exponential function.

Therefore, the function in the first table is the only exponential function.

Theory

### The Graph of an Exponential Function

The graph of a function with a constant rate of change will be a line, since for every step in the $x$-direction, the change in $y$ is the same. For an exponential function, the change in the $y$-direction shows a constant multiplier, leading to a curve-shaped graph.

The $y$-intercept is usually called the initial value, and the factor by which a $y$-value is multiplied to get to the next is called the constant multiplier.

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Exercise

Use the points on the graph to determine if the function is exponential.

Show Solution
Solution

To definitively state that the function is exponential, we must determine that the rate of change shows a constant multiplier. We can do this by finding the rate of change between consecutive points. Let's start with the two left-most points, $(0,16) \quad \text{and} \quad (1,8).$ Since $8$ is half of $16$ we can multiply $16$ by $0.5$ to get $8.$

The function will be exponential if we can multiply each of the $y$-values by $0.5$ to get to the $y$-value of the subsequent point. Does multiplying $8$ by $0.5$ get us to the next point? $8\cdot 0.5 = {\color{#0000FF}{4}}$ Yes! The next coordinate is $(2,{\color{#0000FF}{4}}).$

When taking another step to the right, we should land on $(3,2)$ because $2$ is half of $4.$

For every step in the $x$-direction, the $y$-value changes by a multiplier of $0.5.$ We can assume that this behavior continues for the function that extends infinitely. Thus, the function is exponential.

Concept

## Exponential Function

Formally, any function that can be written in the following form is an exponential function.

$y=a \cdot b^x$

Here, the coefficient, $a,$ is the initial value. The base, $b,$ can be interpreted as the constant multiplier.

Therefore, for all exponential functions $y=a \cdot b^x,$ $a\neq 0$ and $b>0, b\neq 1.$