Pre-Calculus Examples of Graphing Rational Functions with Asymptotes

📚 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 is $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. 💡 Vertical Asymptotes: Occur where the denominator $Q(x) = 0$, but the numerator $P(x) \neq 0$. Solve for $x$ to find the vertical asymptote(s). 📈 Horizontal Asymptotes: Determined by comparing the degrees of the numerator and denominator: * 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 might be a slant asymptote). 📐 Slant (Oblique) Asymptotes: Occur when the degree of $P(x)$ is exactly one greater than the degree of $Q(x)$. Find it by performing polynomial long division; the quotient (without the remainder) is the equation of the slant asymptote. ✏️ Holes: Occur when a factor cancels out from both the numerator and denominator. To find the $x$-coordinate of the hole, set the cancelled factor equal to zero and solve for $x$. To find the $y$-coordinate, plug the $x$-value into the simplified function.

Practice Quiz

1. What is the vertical asymptote of the function $f(x) = \frac{x+2}{x-3}$?
1. A) $x = -2$
2. B) $x = 3$
3. C) $y = 1$
4. D) $y = 0$
2. What is the horizontal asymptote of the function $f(x) = \frac{2x^2 + 1}{x^2 - 4}$?
1. A) $y = 0$
2. B) $y = 1$
3. C) $y = 2$
4. D) No horizontal asymptote
3. Which of the following functions has a hole?
1. A) $f(x) = \frac{x+1}{x-2}$
2. B) $f(x) = \frac{x^2 - 1}{x+1}$
3. C) $f(x) = \frac{x}{x^2+1}$
4. D) $f(x) = \frac{1}{x}$
4. What is the slant asymptote of the function $f(x) = \frac{x^2 + 2x + 1}{x}$?
1. A) $y = x$
2. B) $y = x + 2$
3. C) $y = 2x + 1$
4. D) No slant asymptote
5. For the function $f(x) = \frac{x-4}{(x-4)(x+2)}$, what is the x-coordinate of the hole?
1. A) $x = -2$
2. B) $x = 4$
3. C) $x = 0$
4. D) There is no hole
6. What are the vertical asymptotes of $f(x) = \frac{1}{x^2 - 9}$?
1. A) $x = 3$ only
2. B) $x = -3$ only
3. C) $x = 3$ and $x = -3$
4. D) No vertical asymptotes
7. What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 1}$?
1. A) $y = 5$
2. B) $y = 1$
3. C) $y = 0$
4. D) No horizontal asymptote