#AP Pre-Calculus: Rational Functions & End Behavior 🚀

Hey there, future AP Pre-Calc master! Let's dive into the world of rational functions and their end behavior. This guide is designed to be your go-to resource the night before the exam. Let's make sure you're feeling confident and ready to ace it! 💪

#1.7: Rational Functions and End Behavior

#What are Rational Functions? 🤔

Remember our polynomial friends? Well, rational functions are like their cool cousins! A rational function is simply a ratio (or quotient) of two polynomial functions. Think of it like this:

$$\text{Rational Function} = \frac{\text{Polynomial}}{\text{Polynomial}}$$

Image: A visual of a rational expression, showing a polynomial divided by another polynomial.

#End Behavior: What Happens at the Edges? 🧐

End behavior describes what happens to the function as x approaches positive or negative infinity. We're looking at the degrees of the polynomials in the numerator and denominator to figure this out. The polynomial with the higher degree will have the greatest influence on the overall behavior of the rational function.

Let's break it down:

#(1) Numerator Polynomial Dominates ⬆️

If the degree of the numerator is greater than the degree of the denominator, the rational function's end behavior is determined by the quotient of the leading terms. This quotient will be a non-constant polynomial.

• Slant Asymptotes: If the quotient is a linear polynomial, the rational function will have a slant asymptote. The function will approach this line as x goes to infinity or negative infinity.

Image: A graph showing a rational function with a slant asymptote.

#(2) Neither Polynomial Dominates ↔️

If the degrees of the numerator and denominator are the same, the quotient of the leading terms is a constant. This constant gives us the horizontal asymptote, which the function approaches as x goes to infinity or negative infinity.

#(3) Denominator Polynomial Dominates ⬇️

If the degree of the denominator is greater than the degree of the numerator, the function has a horizontal asymptote at y = 0. As x gets very large, the denominator grows much faster than the numerator, making the overall fraction approach zero.

Image: A graph showing a rational function with a horizontal asymptote at y=0.

#Limits: Formalizing End Behavior 🎯

When a rational function has a horizontal asymptote at y = b, we express this using limits:

$$\lim_{x \to \pm \infty} r(x) = b$$

This means that as x approaches positive or negative infinity, the function r(x) approaches the value b. The function gets arbitrarily close to b without necessarily reaching it.

Image: Two graphs illustrating limits as x approaches infinity.

#Final Exam Focus 📝

• Highest Priority: Understanding the relationship between polynomial degrees and end behavior. Identifying horizontal and slant asymptotes.
• Common Question Types:
  • Matching rational functions to their graphs.
  • Finding asymptotes (horizontal, vertical, and slant).
  • Describing end behavior using limit notation.

#Last-Minute Tips 💡

• Time Management: Start with the easier questions to build confidence. Don't spend too long on one problem; move on and come back if you have time.
• Common Pitfalls: Be careful with signs and exponents. Double-check your work, especially when simplifying rational expressions.
• Challenging Questions: If you're stuck, try sketching a quick graph. Visualizing the function can often help you understand its behavior.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. What is the end behavior of the rational function $f(x) = \frac{3x^3 - 2x + 1}{x^2 + 5}$ as x approaches infinity? (A) Approaches 0 (B) Approaches 3 (C) Approaches infinity (D) Approaches -infinity

2. Which of the following rational functions has a horizontal asymptote at y = 2? (A) $f(x) = \frac{2x}{x^2 + 1}$ (B) $f(x) = \frac{2x^2}{x^2 + 1}$ (C) $f(x) = \frac{x}{2x^2 + 1}$ (D) $f(x) = \frac{x^2}{2x + 1}$

3. The rational function $g(x) = \frac{x^2 + 3x - 4}{x-1}$ has a slant asymptote. What is the equation of the slant asymptote? (A) y = x + 4 (B) y = x - 4 (C) y = x + 2 (D) y = x - 2

#Free Response Question

Consider the rational function $h(x) = \frac{2x^2 - 8}{x^2 - 4x + 3}$.

(a) Find the vertical asymptotes of $h(x)$. (2 points)

(b) Find the horizontal asymptote of $h(x)$. (2 points)

(c) Determine the end behavior of $h(x)$ as $x \to \infty$ and $x \to -\infty$ using limit notation. (2 points)

(d) Sketch a graph of $h(x)$ showing all asymptotes. (3 points)

Scoring Breakdown:

(a) Vertical asymptotes: Setting the denominator equal to zero and solving for x. $x^2 - 4x + 3 = (x-3)(x-1) = 0$. Therefore, x = 1 and x = 3. (1 point for each correct asymptote)

(b) Horizontal asymptote: Since the degree of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. $y = \frac{2}{1} = 2$. (2 points for the correct asymptote)

(c) End behavior using limit notation: $\lim_{x \to \infty} h(x) = 2$ and $\lim_{x \to -\infty} h(x) = 2$. (1 point for each correct limit)

(d) Sketch: Correctly showing asymptotes at x=1, x=3, and y=2. (1 point for each correct asymptote drawn) and a graph of the function approaching asymptotes. (1 point for the shape of the graph)

You've got this! You are well-prepared. Go get that 5! 🎉