- Understand what exponential functions are and learn their main features
- Write the equation for an exponential function
- Draw graphs of exponential functions
- Modify graphs of exponential functions using shifts, stretches, and reflections
Defining Exponential Functions
What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.
- Percent change refers to a change based on a percent of the original amount.
- Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
- Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.
For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth.
| [latex]x[/latex] | [latex]y = 2^x[/latex] | [latex]y = 2x[/latex] |
| [latex]0[/latex] | [latex]y = 2^0 = 1[/latex] | [latex]y = 2(0) = 0[/latex] |
| [latex]1[/latex] | [latex]y = 2^1 = 2[/latex] | [latex]y = 2(1) = 2[/latex] |
| [latex]2[/latex] | [latex]y = 2^2 = 4[/latex] | [latex]y = 2(2) = 4[/latex] |
| [latex]3[/latex] | [latex]y = 2^3 = 8[/latex] | [latex]y = 2(3) = 6[/latex] |
| [latex]4[/latex] | [latex]y = 2^4 = 16[/latex] | [latex]y = 2(4) = 8[/latex] |
We can infer that for these two functions, exponential growth dwarfs linear growth.
- Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
- Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.
Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding [latex]2[/latex] to the output whenever the input was increased by one.
exponential function
The general form of the exponential formula is
[latex]f(x)=ab^x[/latex]
where [latex]a[/latex] is any nonzero number and [latex]b[/latex] is a positive real number not equal to [latex]1[/latex].
- if [latex]b>1[/latex], the function grows at a rate proportional to its size.
- if [latex]0 \lt b \lt 1[/latex], the function decays at a rate proportional to its size.
This is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:
- Consider a base of –9 and exponent of [latex]\frac{1}{2}[/latex]. Then [latex]f\left(x\right)=f\left(\frac{1}{2}\right)={\left(-9\right)}^{\frac{1}{2}}=\sqrt{-9}[/latex], which is not a real number.
Why do we limit the base to positive values other than 1?
This is because a base of 1 results in the constant function. Observe what happens if the base is 1:
- Consider a base of 1. Then [latex]f\left(x\right)={1}^{x}=1[/latex] for any value of x.
To evaluate an exponential function with the form [latex]f\left(x\right)={b}^{x}[/latex], we simply substitute [latex]x[/latex] with the given value, and calculate the resulting power.
[latex]\begin{array}{llllllll}f\left(x\right)\hfill & ={2}^{x}\hfill & \hfill \\ f\left(3\right)\hfill & ={2}^{3}\text{}\hfill & \text{Substitute }x=3. \hfill \\ \hfill & =8\text{}\hfill & \text{Evaluate the power}\text{.}\hfill \end{array}[/latex]
When evaluating an exponential function, it is important to follow the order of operations.
[latex]\begin{array}{c}f\left(x\right)\hfill & =30{\left(2\right)}^{x}\hfill & \hfill \\ f\left(3\right)\hfill & =30{\left(2\right)}^{3}\hfill & \text{Substitute }x=3.\hfill \\ \hfill & =30\left(8\right)\text{ }\hfill & \text{Simplify the power first}\text{.}\hfill \\ \hfill & =240\hfill & \text{Multiply}\text{.}\hfill \end{array}[/latex]
Note that if the order of operations were not followed, the result would be incorrect:
[latex]f\left(3\right)=30{\left(2\right)}^{3}\ne {60}^{3}=216,000[/latex]