# Characteristics of Graphs of Exponential Functions

### Learning Outcomes

- Determine whether an exponential function and its associated graph represents growth or decay.
- Sketch a graph of an exponential function.

x |
–3 | –2 | –1 | 0 | 1 | 2 | 3 |

[latex]f\left(x\right)={2}^{x}[/latex] |
[latex]\frac{1}{8}[/latex] | [latex]\frac{1}{4}[/latex] | [latex]\frac{1}{2}[/latex] | 1 | 2 | 4 | 8 |

*constant ratio*. In fact, for any exponential function with the form [latex]f\left(x\right)=a{b}^{x}[/latex],

*b*is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of

*a*. Notice from the table that:

- the output values are positive for all values of
*x* - as
*x*increases, the output values increase without bound - as
*x*decreases, the output values grow smaller, approaching zero

Notice that the graph gets close to the x-axis but never touches it.

The domain of [latex]f\left(x\right)={2}^{x}[/latex] is all real numbers, the range is [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote is [latex]y=0[/latex].
To get a sense of the behavior of **exponential decay**, we can create a table of values for a function of the form [latex]f\left(x\right)={b}^{x}[/latex] whose base is between zero and one. We’ll use the function [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex]. Observe how the output values in the table below change as the input increases by 1.

x |
–3 | –2 | –1 | 0 | 1 | 2 | 3 |

[latex]g\left(x\right)=\left(\frac{1}{2}\right)^{x}[/latex] |
8 | 4 | 2 | 1 | [latex]\frac{1}{2}[/latex] | [latex]\frac{1}{4}[/latex] | [latex]\frac{1}{8}[/latex] |

- the output values are positive for all values of
*x* - as
*x*increases, the output values grow smaller, approaching zero - as
*x*decreases, the output values grow without bound

The domain of [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex] is all real numbers, the range is [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote is [latex]y=0[/latex].

### A General Note: Characteristics of the Graph of the Parent Function [latex]f\left(x\right)={b}^{x}[/latex]

An exponential function with the form [latex]f\left(x\right)={b}^{x}[/latex], [latex]b>0[/latex], [latex]b\ne 1[/latex], has these characteristics:**one-to-one**function- horizontal asymptote: [latex]y=0[/latex]
- domain: [latex]\left(-\infty , \infty \right)[/latex]
- range: [latex]\left(0,\infty \right)[/latex]
*x-*intercept: none*y-*intercept: [latex]\left(0,1\right)[/latex]- increasing if [latex]b>1[/latex]
- decreasing if [latex]b<1[/latex]

**exponential growth**and

**decay**functions. Try decimal values between [latex]0[/latex] and [latex]1[/latex], and also values greater than [latex]1[/latex]. Which one is growth and which one is decay?

### How To: Given an exponential function of the form [latex]f\left(x\right)={b}^{x}[/latex], graph the function

- Create a table of points.
- Plot at least 3 point from the table including the
*y*-intercept [latex]\left(0,1\right)[/latex]. - Draw a smooth curve through the points.
- State the domain, [latex]\left(-\infty ,\infty \right)[/latex], the range, [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote, [latex]y=0[/latex].

### tip for success

When sketching the graph of an exponential function by plotting points, include a few input values left and right of zero as well as zero itself. With few exceptions, such as functions that would be undefined at zero or negative input like the radical or (as you'll see soon) the logarithmic function, it is good practice to let the input equal [latex]-3, -2, -1, 0, 1, 2, \text{ and } 3[/latex] to get the idea of the shape of the graph.### Example: Sketching the Graph of an Exponential Function of the Form [latex]f\left(x\right)={b}^{x}[/latex]

Sketch a graph of [latex]f\left(x\right)={0.25}^{x}[/latex]. State the domain, range, and asymptote.Answer: Before graphing, identify the behavior and create a table of points for the graph.

- Since
*b*= 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote*y*= 0. - Create a table of points.
**x**–3 –2 –1 0 1 2 3 **[latex]f\left(x\right)={0.25}^{x}[/latex]**64 16 4 1 0.25 0.0625 0.015625 - Plot the
*y*-intercept, [latex]\left(0,1\right)[/latex], along with two other points. We can use [latex]\left(-1,4\right)[/latex] and [latex]\left(1,0.25\right)[/latex].

The domain is [latex]\left(-\infty ,\infty \right)[/latex], the range is [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote is [latex]y=0[/latex].

### Example

Sketch a graph of [latex]f(x)={\sqrt{2}(\sqrt{2})}^{x}[/latex]. State the domain and range.Answer:

Before graphing, identify the behavior and create a table of points for the graph.

- Since
*b*= [latex]\sqrt{2}[/latex], which is greater than one, we know the function is increasing, and we can verify this by creating a table of values. The left tail of the graph will get really close to the x-axis and the right tail will increase without bound. - Create a table of points.
**x**[latex]–3[/latex] [latex]–2[/latex] [latex]–1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] **[latex]f\left(x\right)=\sqrt{2}{(\sqrt{2})}^{x}[/latex]**[latex]0.5[/latex] [latex]0.71[/latex] [latex]1[/latex] [latex]1.41[/latex] [latex]2[/latex] [latex]2.83[/latex] [latex]4[/latex] - Plot the
*y*-intercept, [latex]\left(0,1.41\right)[/latex], along with two other points. We can use [latex]\left(-1,1\right)[/latex] and [latex]\left(1,2\right)[/latex].

Draw a smooth curve connecting the points.

The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex].### Try It

Sketch the graph of [latex]f\left(x\right)={4}^{x}[/latex]. State the domain, range, and asymptote.Answer: The domain is [latex]\left(-\infty ,\infty \right)[/latex], the range is [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote is [latex]y=0[/latex].

[ohm_question]3607[/ohm_question]## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Characteristics of Graphs of Exponential Functions Interactive.
**Authored by:**Lumen Learning.**Located at:**https://www.desmos.com/calculator/3pqyjsunkg.**License:**Public Domain: No Known Copyright. - Graph a Basic Exponential Function Using a Table of Values.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Graph an Exponential Function Using a Table of Values.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Question ID 3607.
**Provided by:**Reidel,Jessica**Authored by:**Jay Abramson, et al..**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL. - College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected]. - Precalculus.
**Provided by:**OpenStax**Authored by:**Jay Abramson, et al..**Located at:**https://openstax.org/books/precalculus/pages/1-introduction-to-functions.**License:**CC BY: Attribution.**License terms:**Download For Free at : http://cnx.org/contents/[email protected]..