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Graphing Rational Functions

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Graphing Rational Functions

Study Guide

Key Definition

A rational function is a function of the form $y = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials.

Important Notes

Asymptotes are lines that the graph of the function approaches but never touches.
Vertical asymptotes occur where the denominator is zero.
Horizontal asymptotes are determined by the degrees of the polynomials.
Intercepts are points where the graph crosses the axes.
The graph of a rational function can have multiple asymptotes.

Mathematical Notation

$\frac{p(x)}{q(x)}$ denotes a rational function

$x^n$ denotes an exponent

$\deg p(x)$ denotes the degree of polynomial p(x)

Remember to use proper notation when solving problems

Why It Works

Understanding the relationship between the numerator and denominator polynomials helps in identifying asymptotes and intercepts in $rational functions$.

Remember

Always check for vertical asymptotes by setting the denominator equal to zero and solving for $x$.

Quick Reference

Vertical Asymptote:$Solve\;q(x)=0\;(with\;p(x)\neq0)$

Horizontal Asymptote:$\deg p < \deg q \Rightarrow y=0;\;\deg p=\deg q \Rightarrow y=\dfrac{\text{leading coeff. of }p}{\text{leading coeff. of }q};\;\deg p>\deg q \Rightarrow \text{no horizontal asymptote}$

Understanding Graphing Rational Functions

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A rational function is the quotient of two polynomials, for example $y=\frac{1}{x-2}$. To graph it, find the x-intercepts by setting the numerator to zero and the vertical asymptotes by setting the denominator to zero. Plot a few points and sketch the curve approaching asymptotes.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the vertical asymptote of the function $y = \frac{1}{x - 2}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder is designing a ramp with a rational function $f(x) = \frac{3x}{x^2 - 1}$. Find the vertical asymptotes to ensure safety.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the function $y = \frac{x^2 + 1}{x - 3}$. What happens to the graph as $x$ approaches 3?

Challenge Quiz

Single Choice Quiz

Advanced

Determine the horizontal asymptote of the function $y = \frac{2x^2 + 3x + 1}{x^2 - 4}$.

Recap

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Review key concepts and takeaways