john_knight
john_knight 6d ago • 10 views

Rational functions test pdf

Hey there! 👋 Let's ace that rational functions test! Here's a quick study guide to refresh your memory, followed by a practice quiz to boost your confidence. Good luck! 🍀
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theresalewis1989 Dec 29, 2025

📚 Quick Study Guide

  • 🔍 A rational function is a function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials.
  • ➗ The general form of a rational function is $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
  • 📈 Vertical asymptotes occur at $x$ values where the denominator $Q(x) = 0$, provided the numerator $P(x)$ is not also zero at that point.
  • ➖ Holes occur when both the numerator and denominator are zero at the same $x$ value. Simplify the rational function to remove the common factor causing the hole.
  • ↔️ Horizontal asymptotes are determined by comparing the degrees of $P(x)$ and $Q(x)$:
    • If degree of $P(x)$ < degree of $Q(x)$, then $y = 0$ is the horizontal asymptote.
    • If degree of $P(x)$ = degree of $Q(x)$, then $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$ is the horizontal asymptote.
    • If degree of $P(x)$ > degree of $Q(x)$, then there is no horizontal asymptote (but there may be a slant asymptote).
  • ✏️ To find the $x$-intercepts, set $P(x) = 0$ and solve for $x$.
  • 📍 To find the $y$-intercept, evaluate $f(0)$, i.e., $f(0) = \frac{P(0)}{Q(0)}$.

🧪 Practice Quiz

  1. What is the vertical asymptote of the rational function $f(x) = \frac{x+2}{x-3}$?
    1. x = -2
    2. x = 2
    3. x = -3
    4. x = 3
  2. What is the horizontal asymptote of the rational function $f(x) = \frac{2x^2 + 1}{x^2 - 4}$?
    1. y = 0
    2. y = 1
    3. y = 2
    4. No horizontal asymptote
  3. Which of the following rational functions has a hole at $x = 1$?
    1. $f(x) = \frac{x+1}{x-1}$
    2. $f(x) = \frac{x-1}{x+1}$
    3. $f(x) = \frac{(x-1)(x+2)}{x-1}$
    4. $f(x) = \frac{x+2}{x-1}$
  4. What is the $x$-intercept of the rational function $f(x) = \frac{x-5}{x+2}$?
    1. x = -2
    2. x = 2
    3. x = -5
    4. x = 5
  5. What is the $y$-intercept of the rational function $f(x) = \frac{x+3}{x-1}$?
    1. y = -3
    2. y = 3
    3. y = -1
    4. y = -3
  6. What is the domain of the rational function $f(x) = \frac{1}{x^2-9}$?
    1. All real numbers
    2. $x \neq 3$
    3. $x \neq -3$
    4. $x \neq 3$ and $x \neq -3$
  7. What is the oblique (slant) asymptote of the rational function $f(x) = \frac{x^2+2x+1}{x-1}$?
    1. y = x + 1
    2. y = x + 3
    3. y = x - 1
    4. There is no oblique asymptote
Click to see Answers
  1. D
  2. C
  3. C
  4. D
  5. A
  6. D
  7. B

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