How to Put A Horizantal Asymptote on A Graphing Calculator

Horizontal asymptotes are essential in graphing rational functions. They help determine the behavior of the graph as x approaches positive or negative infinity. This guide explains how to identify and graph horizontal asymptotes using a graphing calculator.

What is a Horizontal Asymptote?

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity. For rational functions (fractions where both the numerator and denominator are polynomials), horizontal asymptotes can be found by comparing the degrees of the numerator and denominator.

Formula: For a rational function f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials:

If degree of P(x) < degree of Q(x): Horizontal asymptote is y = 0
If degree of P(x) = degree of Q(x): Horizontal asymptote is y = (leading coefficient of P(x))/(leading coefficient of Q(x))
If degree of P(x) > degree of Q(x): There is no horizontal asymptote (may have an oblique asymptote)

How to Graph Horizontal Asymptotes

Graphing horizontal asymptotes involves several steps:

Identify the horizontal asymptote using the formula above
Plot the horizontal asymptote as a dashed line on your graph
Graph the function itself, ensuring it approaches the asymptote but never touches it
Label the asymptote with its equation

Note: Horizontal asymptotes are never touched by the graph of the function. They represent the function's behavior at infinity.

Calculator Methods

Most graphing calculators can help you identify and graph horizontal asymptotes. Here are methods for common calculators:

TI-84 Graphing Calculator

Enter the function in Y=
Set the window to show the relevant portion of the graph
Graph the function (press GRAPH)
Use the TRACE feature to identify where the graph approaches a horizontal line
Use the TABLE feature to check values as x approaches infinity

Desmos Calculator

Enter the function in the input box
Zoom out to see the behavior at infinity
Desmos will automatically show horizontal asymptotes when they exist
You can manually add a dashed line for the asymptote if needed

Graphing.com

Enter the function in the input field
Adjust the x-range to see the behavior at infinity
Graphing.com will display horizontal asymptotes automatically
You can add a horizontal line manually if needed

Example

Let's graph the function f(x) = (3x² + 2)/(2x² + 5) and find its horizontal asymptote.

Step 1: Compare degrees of numerator and denominator

Numerator: 3x² + 2 (degree 2)

Denominator: 2x² + 5 (degree 2)

Since degrees are equal, horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator)

y = 3/2 = 1.5

Using a graphing calculator:

Enter the function in Y=
Set the window to show x from -10 to 10 and y from 0 to 2
Graph the function and observe it approaches y=1.5 as x approaches ±∞
Add a dashed horizontal line at y=1.5 to represent the asymptote

FAQ

What if a function has no horizontal asymptote?: If the degree of the numerator is greater than the denominator, there will be no horizontal asymptote. The function may have an oblique (slant) asymptote instead.
Can a function have more than one horizontal asymptote?: No, a function can have at most one horizontal asymptote. It represents the behavior of the function as x approaches positive or negative infinity.
How do I know if my graphing calculator is showing the correct asymptote?: Check the behavior of the function at very large positive and negative x-values. The graph should approach the horizontal asymptote but never touch it.
What if my function has a hole where it would intersect the horizontal asymptote?: If the function has a hole (removable discontinuity) at the point where it would intersect the horizontal asymptote, the graph will not touch the asymptote at that point.