Understanding Horizontal Asymptotes of Rational Functions | Precalculus

What are Horizontal Asymptotes?

• 📈 Horizontal asymptotes describe the end behavior of a function as x approaches positive or negative infinity.
• 💡 They represent a horizontal line that the function's graph approaches but may not always touch.
• ⚠️ The horizontal asymptote is not always the x-axis; it can be any horizontal line.

Analyzing Rational Functions

• ➗ A rational function is expressed as a ratio of two polynomials.
• 🔑 To find the horizontal asymptote, focus on the highest degree terms in the numerator and the denominator.
• 🚀 As x becomes very large (positive or negative), these dominant terms dictate the function's behavior.

Method for Finding Horizontal Asymptotes

• 📊 For a rational function $f(x) = \frac{P(x)}{Q(x)}$, compare the degrees of the highest degree terms in $P(x)$ and $Q(x)$.
• 💡 If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of these terms.
• 📉 If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $y=0$ (the x-axis).
• 📈 If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Example 1: Equal Degrees

• 🎯 Consider $f(x) = \frac{-5x^2 - 6x + 4}{2x^2 + 8}$.
• 🔑 The highest degree terms are $-5x^2$ and $2x^2$.
• ✅ The horizontal asymptote is $y = \frac{-5}{2}$ or $y = -2.5$.

Example 2: Degree of Denominator Greater

• 🎯 Consider $g(x) = \frac{(x - 4)^2}{x^2 + 5 * (3x - 2)}$.
• 🔑 The highest degree term in the numerator is $x^2$ (from $(x-4)^2$).
• 🔑 The highest degree term in the denominator is $3x^3$ (from $x^2 * 3x$).
• ✅ Since the degree of the denominator (3) is greater than the degree of the numerator (2), the horizontal asymptote is $y=0$.