Learn on PengienVision, Algebra 2Chapter 4: Rational Functions

Lesson 2: Graphing Rational Functions

In this Grade 11 enVision Algebra 2 lesson, students learn how to graph rational functions by identifying vertical and horizontal asymptotes and rewriting rational expressions using polynomial long division. The lesson covers key techniques such as expressing a rational function in the form a/(x-h) + k to reveal transformations of the parent function f(x) = 1/x, and applying degree comparison rules to determine horizontal asymptote behavior. Students also practice factoring denominators to locate vertical asymptotes and sketch accurate graphs of rational functions.

Section 1

Rational Functions

Property

A rational function is a function defined by an algebraic fraction. That is, it has the form

f(x) = \frac{P(x)}{Q(x)}

where $P(x)$ and $Q(x)$ are polynomials. A rational function is undefined at any $x$ -values where $Q(x) = 0$ . A vertical asymptote is a vertical line on the graph that occurs where a rational function is undefined.

Examples

The function $f(x) = \frac{20}{x-5}$ is a rational function. It is undefined when $x=5$ , so its graph has a vertical asymptote at the line $x=5$ .
For the function $g(x) = \frac{x+1}{x^2-9}$ , the denominator is zero when $x=3$ or $x=-3$ . The graph has two vertical asymptotes: $x=3$ and $x=-3$ .
The function for average cost $C(n) = \frac{200+8n}{n}$ is rational. It is undefined for $n=0$ , which means you cannot produce zero items and calculate a meaningful average cost.

Explanation

A rational function is just a fraction made of polynomials. Its graph has a special feature called a vertical asymptote—a vertical line the graph gets very close to but never crosses. This line appears wherever the denominator is zero.

Section 2

Reciprocal Functions and Asymptotes

Property

The basic reciprocal functions are $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x^2}$ . An asymptote is a line that the graph of a function approaches but never touches. For these functions, the y-axis ( $x=0$ ) is a vertical asymptote, and the x-axis ( $y=0$ ) is a horizontal asymptote.

Examples

For $f(x) = \frac{1}{x}$ , as $x$ gets very large, like $x=1000$ , $f(x)$ becomes very small, $f(1000) = 0.001$ . This shows the graph approaching the horizontal asymptote $y=0$ .
For $g(x) = \frac{1}{x^2}$ , as $x$ approaches $0$ from either side, like $x=0.1$ or $x=-0.1$ , $g(x)$ becomes a large positive number, $g(0.1)=100$ . This shows the vertical asymptote at $x=0$ .
The function $f(x) = \frac{1}{x}$ is undefined at $x=0$ because division by zero is not allowed. This is why the graph exists as two separate branches and never crosses the y-axis.

Explanation

These graphs show what happens when you divide 1 by a number. As the number gets close to zero, the result shoots towards infinity. As the number gets huge, the result shrinks towards zero, but never quite reaches it.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Rational Functions

Property

A rational function is a function defined by an algebraic fraction. That is, it has the form

f(x) = \frac{P(x)}{Q(x)}

Examples

The function $f(x) = \frac{20}{x-5}$ is a rational function. It is undefined when $x=5$ , so its graph has a vertical asymptote at the line $x=5$ .
For the function $g(x) = \frac{x+1}{x^2-9}$ , the denominator is zero when $x=3$ or $x=-3$ . The graph has two vertical asymptotes: $x=3$ and $x=-3$ .
The function for average cost $C(n) = \frac{200+8n}{n}$ is rational. It is undefined for $n=0$ , which means you cannot produce zero items and calculate a meaningful average cost.

Explanation

Section 2

Reciprocal Functions and Asymptotes

Property

Examples

For $f(x) = \frac{1}{x}$ , as $x$ gets very large, like $x=1000$ , $f(x)$ becomes very small, $f(1000) = 0.001$ . This shows the graph approaching the horizontal asymptote $y=0$ .
For $g(x) = \frac{1}{x^2}$ , as $x$ approaches $0$ from either side, like $x=0.1$ or $x=-0.1$ , $g(x)$ becomes a large positive number, $g(0.1)=100$ . This shows the vertical asymptote at $x=0$ .
The function $f(x) = \frac{1}{x}$ is undefined at $x=0$ because division by zero is not allowed. This is why the graph exists as two separate branches and never crosses the y-axis.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Rational Functions

Property

A rational function is a function defined by an algebraic fraction. That is, it has the form

f(x) = \frac{P(x)}{Q(x)}

Examples

The function $f(x) = \frac{20}{x-5}$ is a rational function. It is undefined when $x=5$ , so its graph has a vertical asymptote at the line $x=5$ .
For the function $g(x) = \frac{x+1}{x^2-9}$ , the denominator is zero when $x=3$ or $x=-3$ . The graph has two vertical asymptotes: $x=3$ and $x=-3$ .
The function for average cost $C(n) = \frac{200+8n}{n}$ is rational. It is undefined for $n=0$ , which means you cannot produce zero items and calculate a meaningful average cost.

Explanation

Section 2

Reciprocal Functions and Asymptotes

Property

Examples

For $f(x) = \frac{1}{x}$ , as $x$ gets very large, like $x=1000$ , $f(x)$ becomes very small, $f(1000) = 0.001$ . This shows the graph approaching the horizontal asymptote $y=0$ .
For $g(x) = \frac{1}{x^2}$ , as $x$ approaches $0$ from either side, like $x=0.1$ or $x=-0.1$ , $g(x)$ becomes a large positive number, $g(0.1)=100$ . This shows the vertical asymptote at $x=0$ .
The function $f(x) = \frac{1}{x}$ is undefined at $x=0$ because division by zero is not allowed. This is why the graph exists as two separate branches and never crosses the y-axis.