Graph of Rational Functions

Principle of Graphing

A rational function is a function that can be expressed as the ratio of two polynomials. It has the form:

$$f(x) = \frac{P(x)}{Q(x)}$$

Steps to Graph a Rational Function

  1. Identify the Domain: The domain of a rational function is all real numbers except where the denominator \(Q(x)\) is zero. Find the values of \(x\) that make \(Q(x) = 0\).
  2. Find the Intercepts:
    • Y-intercept: Set \(x = 0\) and solve for \(f(0)\).
    • X-intercept: Set \(P(x) = 0\) and solve for \(x\).
  3. Determine Asymptotes:
    • Vertical Asymptotes: These occur where \(Q(x) = 0\). Solve \(Q(x) = 0\) to find the vertical asymptotes.
    • Horizontal Asymptotes: Compare the degrees of \(P(x)\) and \(Q(x)\):
      • 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 (but there may be an oblique asymptote).
  4. Analyze the Behavior Near Asymptotes: Determine how the function behaves as it approaches the vertical and horizontal asymptotes.
  5. Plot Points: Choose values of \(x\) and calculate \(f(x)\) to get points on the graph. Plot these points.
  6. Draw the Graph: Use the information from the intercepts, asymptotes, and plotted points to sketch the graph.

Example

Let's graph the rational function: \[f(x) = \frac{x^2 - 4}{x - 2}\]

  1. Domain:
    • The denominator \(x - 2 = 0\) when \(x = 2\). So, the domain is \(x \ne 0\).
  2. Intercepts:
    • Y-intercept:
      Set \(x = 0\) \[f(0) = \frac{0^2 - 4}{0 - 2}\] \[= \frac{- 4}{- 2}\] \[= 2\] So, the y-intercept is \((0, 2)\).
    • X-intercept:
      Set \[x^2 - 4 = 0:\] \[ ⟹(x - 2)(x + 2) = 0:\] So, the x-intercepts are \(x = -2 \) and \(x = 2 \).
  3. Asymptotes:
    • Vertical Asymptote: \[x = 2\].
    • Horizontal Asymptote:
      The degree of the numerator (2) is greater than the degree of the denominator (1), so there is no horizontal asymptote.
  4. Behavior Near Asymptotes:
    • As \(x\)approaches 2 from the left, \(f(x)\) tends to ∞.
    • As \(x\) approaches 2 from the right, \(f(x)\) tends to − ∞.
  5. Plot Points:
    • Choose values of \(x\) such as \(x\) = - 3, -1, 1, 3 and calculate \(f(x)\).
  6. Draw the Graph:
    • Sketch the graph using the intercepts, asymptotes, and plotted points.

Follow the steps above to graph this function.

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