Rational Functions Notes

Graph of a Rational Function

• The graph of $y = \frac{f(x)}{g(x)} = \frac{a<em>n x^n + …}{b</em>m x^m + …}$ has the following characteristics:

1. End Behavior Asymptote:

• A. If n < m:
  • The end behavior asymptote is the horizontal asymptote $y = 0$.
• B. If n = m:
  • The end behavior asymptote is the horizontal asymptote $y = \frac{a<em>n}{b</em>m}$, which is the ratio of leading coefficients.
• C. If n > m:
  • The end behavior asymptote is the quotient polynomial function $y = q(x)$, where $f(x) = q(x) \cdot g(x) + r(x)$.
  • There is no horizontal asymptote.
  • If $n - m = 1$, $y = q(x)$ is a slant asymptote.

2. x-intercepts:

• These occur at the real zeros of the numerator that are not also zeros of the denominator.

3. y-intercept:

• This is the value of $f(0)$, if defined.

4. Vertical Asymptote and Hole:

• For $y = \frac{(x-a)^n}{(x-a)^m}$
  • A. If n < m:
    • There will be a vertical asymptote at $x = a$ (Non-Removable Discontinuity).
  • B. If n >= m:
    • There will be a hole at $x = a$ (Removable Discontinuity).
    • The y-value of the hole can be found by reducing the factors and substituting $x = a$ in the reduced function.

Discontinuities:

• Continuous
• Hole
• Jump
• Vertical Asymptote

Example 1: Analyze $f(x) = \frac{1}{x}$:

• A. Find the domain:
  • $x \neq 0$
  • $x \in (-\infty, 0) \cup (0, \infty)$
• B. Find the vertical asymptote:
  • $x = 0$ (y-axis)
• C. Find the range:
  • $y \neq 0$
  • $y \in (-\infty, 0) \cup (0, \infty)$
• D. Find the horizontal asymptote:
  • $y = 0$ (x-axis)
• E. Graph the function and label sample points:
  • Sample points: $(-2, -\frac{1}{2}), (-1, -1), (-\frac{1}{2}, -2), (\frac{1}{2}, 2), (1, 1), (2, \frac{1}{2})$

Example 2: Analyze $f(x) = \frac{x+2}{x^2+3x+2}$:

• A. Rewrite in factored form:
  • $f(x) = \frac{x+2}{(x+2)(x+1)}$
• B. Find the domain:
  • $x \neq -2, x \neq -1$
  • $x \in (-\infty, -2) \cup (-2, -1) \cup (-1, \infty)$
• C. Find the vertical asymptote:
  • $x = -1$
• D. Find the hole:
  • Hole occurs at $x = -2$
  • $f(-2) = \frac{1}{-2+1} = -1$
  • Hole: $(-2, -1)$
• E. Find the horizontal asymptote:
  • $y = 0$
  • Degree of numerator (1) < Degree of denominator (2)
• F. Find the range:
  • $y \in (-\infty, -1) \cup (-1, 0) \cup (0, \infty)$
• G. Find the y-intercept:
  • $f(0) = \frac{0+2}{0+0+2} = 1$
  • $(0, 1)$
• H. Graph the function and label sample points:
  • Note: No x-intercept here.
  • Sample points: $(-4, -\frac{1}{2}), (-3, -\frac{1}{1}), (-\frac{3}{2}, 2), (\frac{3}{2}, \frac{2}{5}), (0, 1), (1, \frac{3}{6}), (2, \frac{1}{2})$

Example 3: Graph and analyze $f(x) = \frac{2x^2-2}{x^2-4}$:

• A. Rewrite in factored form:
  • $f(x) = \frac{2(x^2-1)}{(x+2)(x-2)} = \frac{2(x+1)(x-1)}{(x+2)(x-2)}$
• B. Find the domain:
  • $x \neq -2, x \neq 2$
  • $x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty)$
• C. Find the vertical asymptotes:
  • $x = -2, x = 2$
• D. Find any holes:
  • There are no holes.
• E. Find the horizontal asymptote:
  • $y = 2$
  • Degree of numerator = Degree of denominator, so $y = \frac{2}{1} = 2$ (ratio of leading coefficients).
• F. Find the y-intercept:
  • $f(0) = \frac{-2}{-4} = \frac{1}{2}$
  • $(0, \frac{1}{2})$
• G. Find the x-intercepts:
  • $x+1=0, x-1=0$
  • $x = -1, x = 1$
  • $(-1, 0), (1, 0)$
• H. Use a calculator or Desmos to graph the function. Label sample points:
  • Sample Points: $(-4, 5), (-1, 0), (0, \frac{1}{2}), (1, 0), (4, 5)$
• I. Find the range:
  • $y \in (-\infty, \frac{1}{2}] \cup (2, \infty)$
• J. Describe the end behaviors:
  • Left End Behavior:
    • As $x \to -\infty$, $f(x) \to 2$
    • $\lim_{x \to -\infty} f(x) = 2$
  • Right End Behavior:
    • As $x \to \infty$, $f(x) \to 2$
    • $\lim_{x \to \infty} f(x) = 2$
• K. Describe the behavior near the vertical asymptotes:
  • As $x \to -2^-$, $f(x) \to \infty$
    • $\lim_{x \to -2^-} f(x) = \infty$
  • As $x \to -2^+$, $f(x) \to -\infty$
    • $\lim_{x \to -2^+} f(x) = -\infty$
  • As $x \to 2^-$, $f(x) \to -\infty$
    • $\lim_{x \to 2^-} f(x) = -\infty$
  • As $x \to 2^+$, $f(x) \to \infty$
    • $\lim_{x \to 2^+} f(x) = \infty$

Example 4: Graph and analyze $f(x) = \frac{x^2 - 3x - 4}{x - 2}$:

• A. Rewrite in factored form:
  • $f(x) = \frac{(x+1)(x-4)}{x-2}$
• B. Find the domain:
  • $x \neq 2$
  • $x \in (-\infty, 2) \cup (2, \infty)$
• C. Find the vertical asymptote:
  • $x = 2$
• D. Find any holes:
  • No hole
• E. Find the horizontal asymptote:
  • No horizontal asymptote.
  • Degree of Numerator = 2 > Degree of Denominator = 1
• F. Find the y-intercept:
  • $f(0) = \frac{-4}{-2} = 2$
• G. Find the x-intercepts:
  • $(-1, 0), (4, 0)$
• H. Find the equation of the slant asymptote:
  • Slant Asymptote: $y = x - 1$
• I. Use a calculator or Desmos to graph the function. Label sample points:
  • Sample points: $(-4, -4), (-1, 0), (0, 2), (1, 6), (3, -4), (4, 0), (5, 2), (8, 6)$
• J. Find the range:
  • $y \in (-\infty, \infty)$
• K. Describe the end behaviors:
  • Left End Behavior:
    • As $x \to -\infty$, $f(x) \to -\infty$
    • $\lim_{x \to -\infty} f(x) = -\infty$
  • Right End Behavior:
    • As $x \to \infty$, $f(x) \to \infty$
    • $\lim_{x \to \infty} f(x) = \infty$
• L. Describe the behavior of the function near the vertical asymptote:
  • As $x \to 2^-$, $f(x) \to -\infty$
    • $\lim_{x \to 2^-} f(x) = -\infty$
  • As $x \to 2^+$, $f(x) \to \infty$
    • $\lim_{x \to 2^+} f(x) = \infty$