What is the horizontal asymptote of f(x) = \frac{3x^2 - 5}{x^2 + 2}?

y = 3 The degrees are equal (n=2, m=2). The HA is the ratio of the leading coefficients: 3/1 = 3.

Rational Function Asymptotes

Explore vertical, horizontal, and slant asymptotes of rational functions. Switch between preset functions or enter custom numerator/denominator coefficients to see asymptote detection in real time.

CONCEPT: RATIONAL BEHAVIOR

A rational function is a ratio of two polynomials . Asymptotes describe the behavior of these functions near excluded values (vertical) or at the extreme ends of the graph (horizontal/slant). Understanding these boundaries is essential for sketching accurate graphs and modeling real-world constraints.

MECHANISM: ASYMPTOTE DETECTION

1. **Vertical (VA)**: Occurs where the denominator after simplifying the fraction. If a factor cancels out, it creates a **hole (removable discontinuity)** instead. 2. **Horizontal (HA)**: Compare degrees. If degree(top) < degree(bottom), . If degrees are equal, (ratio of leading coefficients). 3. **Slant (SA)**: Occurs if degree(top) is exactly one higher than degree(bottom). Found using polynomial long division.

HOW TO USE THIS VISUALIZATION

1. **Enter Coefficients**: Modify the values for the numerator and denominator. 2. **Observe the Trace**: Move the cursor along the curve. See how the function value "blows up" toward as it approaches a vertical asymptote. 3. **Toggle Simplification**: See how factoring and canceling terms changes an asymptote into a hole on the graph.

CORE FORMULAS

General rational form

Horizontal asymptote (if n = m)

AP EXAM CONNECTION

Unit: Unit 1: Polynomial and Rational Functions (Topic 1.7-1.11)
Learning Objective: LO 1.7.A

COMMON MISCONCEPTIONS

Thinking a function can never cross an asymptote (functions CAN cross horizontal/slant asymptotes).
Confusing holes with vertical asymptotes.
Assuming every rational function has a horizontal asymptote.