Graphing Rational Functions - January 14, 2014

Today, we delved into the fascinating world of graphing rational functions. Mastering this skill involves understanding a few key concepts that help us accurately sketch these functions. Remember, practice makes perfect, so don't be discouraged if it seems tricky at first! We'll focus on identifying key features of rational functions such as asymptotes and intercepts.

Key Concepts

  • Rational Function Definition: A rational function is defined as a function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
  • Vertical Asymptotes: These occur where the denominator $Q(x) = 0$, provided that the factor does not cancel with a factor in the numerator $P(x)$. For example, in $f(x) = \frac{3}{x-2}$, there's a vertical asymptote at $x=2$.
  • Horizontal Asymptotes: Determined by comparing the degrees of the numerator and denominator:
    • If the degree of $P(x)$ < degree of $Q(x)$, the horizontal asymptote is $y = 0$.
    • If the degree of $P(x)$ = degree of $Q(x)$, the horizontal asymptote is $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$.
    • If the degree of $P(x)$ > degree of $Q(x)$, there is no horizontal asymptote (there might be a slant/oblique asymptote).
  • X-Intercepts: These are the points where $f(x) = 0$, which occurs when $P(x) = 0$ and $Q(x) \neq 0$.
  • Y-Intercepts: This is the point where the graph intersects the y-axis, found by evaluating $f(0)$.

Examples

Let's look at some examples to clarify these concepts:

  1. Example 1: $f(x) = \frac{3}{x-2}$
    • Vertical Asymptote: $x = 2$
    • Horizontal Asymptote: $y = 0$ (degree of numerator < degree of denominator)
    • X-Intercept: None (numerator is a constant)
    • Y-Intercept: $f(0) = \frac{3}{0-2} = -\frac{3}{2}$
  2. Example 2: $f(x) = \frac{2x-1}{x}$
    • Vertical Asymptote: $x = 0$
    • Horizontal Asymptote: $y = 2$ (degrees of numerator and denominator are equal)
    • X-Intercept: $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$
    • Y-Intercept: None (undefined at $x=0$)
  3. Example 3: $f(x) = \frac{x}{x^2 - x - 2} = \frac{x}{(x-2)(x+1)}$
    • Vertical Asymptotes: $x = 2$ and $x = -1$
    • Horizontal Asymptote: $y = 0$
    • X-Intercept: $x = 0$
    • Y-Intercept: $f(0) = \frac{0}{0-0-2} = 0$
  4. Example 4: $f(x) = \frac{x^2}{x^2 + 6x + 9} = \frac{x^2}{(x+3)^2}$
    • Vertical Asymptote: $x = -3$
    • Horizontal Asymptote: $y = 1$
    • X-Intercept: $x = 0$
    • Y-Intercept: $f(0) = \frac{0}{0+0+9} = 0$

Homework

For practice, complete the following problems from the textbook:

  • Page 157: #10-24 even

Keep practicing, and you'll become a rational function graphing pro in no time! Remember to always check for those asymptotes and intercepts – they're your best friends when sketching these graphs.