Characteristics of Graphs of Exponential Functions
Learning Objectives
- Determine whether an exponential function and it's 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 |
- the output values are positive for all values of x;
- as x increases, the output values increase without bound; and
- 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; and
- 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 f(x) = bx
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]
How To: Given an exponential function of the form [latex]f\left(x\right)={b}^{x}[/latex], graph the function by hand.
- 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].
Example: Sketching the Graph of an Exponential Function of the Form f(x) = bx
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]; the horizontal asymptote is [latex]y=0[/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]; the horizontal asymptote is [latex]y=0[/latex].
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.
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].