Chapter 4 Part 3: Rational Functions - Unveiled!

Welcome back to Professor Baker's Math Class! Today, we're conquering rational functions. Get ready to explore their unique properties and master the art of graphing them.

Key Topics Covered:

  • Domain of a Rational Function: We express the domain using interval notation. Remember, the domain excludes any values that make the denominator zero. For example, if $f(x) = \frac{1}{x-2}$, the domain is $(-\infty, 2) \cup (2, \infty)$.
  • Finding Asymptotes:
    • Constant over Linear: Functions like $f(x) = \frac{k}{ax+b}$ have a vertical asymptote at $x = -\frac{b}{a}$ and a horizontal asymptote at $y = 0$ if the degree of the numerator is less than the degree of the denominator.
    • Linear over Linear: For $f(x) = \frac{ax+b}{cx+d}$, the vertical asymptote is at $x = -\frac{d}{c}$ and the horizontal asymptote is at $y = \frac{a}{c}$.
    • Quadratic Numerator or Denominator: Requires factoring and careful analysis. Consider the function $f(x) = \frac{x^2 + 3x - 4}{-(x^2 + 3x + 2)}$. Factoring gives us $f(x) = \frac{(x+4)(x-1)}{-(x+2)(x+1)}$. Vertical asymptotes occur where the denominator is zero, i.e. $x \neq -2$ and $x \neq -1$. The horizontal asymptote is found by examining the leading coefficients: $y = \frac{1}{-1} = -1$.
    • Quadratic over Linear: These often have slant asymptotes, found by polynomial division. For instance, if you have $f(x) = \frac{-x^2 - 5x + 2}{x+3}$, using synthetic division you can find a slant asymptote of $y = -x - 2$. The vertical asymptote is x = -3.
  • Graphing Rational Functions: Practice makes perfect! Remember to find asymptotes, intercepts, and test points in each interval.
  • Graphing Rational Functions with Holes: When a factor cancels out from both the numerator and denominator, you have a hole. For example, consider $g(x) = \frac{5x+25}{x+5}$. This simplifies to $g(x) = \frac{5(x+5)}{x+5} = 5$. There is a hole at x = -5.
  • Matching Graphs with Equations: Look for key features: asymptotes, intercepts, and general shape.
  • Writing Equations from Graphs: Use the asymptotes and intercepts to build the rational function. If you know the graph has x-intercepts at 3 and -1, and vertical asymptotes at x=1 and x=5, a potential function is $f(x) = \frac{a(x-3)(x+1)}{(x-1)(x-5)}$. If you know the graph also passes through (2,2), you can solve for a and completely define f(x).
  • Finding x- and y-intercepts: Set $y = 0$ to find x-intercepts and $x = 0$ to find y-intercepts.
  • Dealing with Multiple Vertical Asymptotes: Identify all values that make the denominator zero. For instance, if $f(x) = \frac{10x-20}{x^2-x-20}$, by factoring we find vertical asymptotes at x=-4 and x=5.

Don't be afraid to experiment and practice! Rational functions can seem tricky at first, but with a solid understanding of these concepts, you'll be graphing like a pro in no time. Keep up the great work!