Analyzing Rational Functions

Study Notes

Steps to Analyze Rational Functions

Simplify the function by canceling any common factors.
Identify X-intercepts by setting the numerator equal to zero and solving for X.
Determine the Y-intercept by substituting X with zero in the function.
Locate vertical asymptotes by setting the denominator equal to zero and solving for X.
Assess horizontal asymptotes based on the degrees of the numerator and denominator:
- No horizontal asymptote if the numerator's degree is higher.
- Horizontal asymptote at y=0 if the denominator's degree is higher.
- If degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
Discover slant asymptotes which occur only when the numerator's degree is one greater than the denominator's; use synthetic division to find their equation.
Check for holes in the graph that occur at canceled factors in the rational function.

X-Intercepts

Found by setting the numerator equal to zero and solving for the respective values of X.

Y-Intercepts

Determined by evaluating the function when X is set to zero.

Vertical Asymptotes

Established by solving the equation obtained from setting the denominator equal to zero.

Horizontal Asymptotes

Baseline rules for identifying horizontal asymptotes depend on the relative degrees of the polynomials involved.

Slant Asymptotes

Unique to situations where the numerator’s degree is precisely one higher than the denominator’s, applicable only when performing synthetic division.

Description

This quiz covers the steps to analyze rational functions in algebra. You will learn how to simplify rational expressions, find X and Y intercepts, identify asymptotes, and check for holes in the graph. Perfect for students looking to master rational function analysis.