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Rational Functions

Partial Fractions
  • To split an improper fraction into partials, calculate the degree of the quotient and work out by comparing coefficients
  • deg(Q(x)) = deg(N(x)) – deg(D(x))
  • where:
    • Q(x) is the quotient
    • N(x) is the numerator
    • D(x) is the denominator
Vertical Asymptote
  • Making the denominator 0 resulting in ∞
  • Example:
    • y = 1/(x + 1)(x – 3)
    • Thus vertical asymptotes at: x = -1 and 3
Horizontal Asymptote
  • By dividing the top and bottom of a fraction by x, we can see what value y tends to when x becomes very large
  • Example:
    • y = 3x – 2/x + 1
    • Divide the numerator and denominator by x
      • y = 3x – 2/x ÷ x + 1/x = 3 – 2/x ÷ 1 + 1/x
    • When x is very large, y = 3/1 = 3
    • Thus the horizontal asymptote at: y = 3
Oblique Asymptotes
  • These occur only with improper fraction
  • Example:
    • y = 2x – 1 + 2/x – 13/x + 2
    • When x becomes very large, y ≈ 2x – 1
    • Thus oblique asymptote at: y = 2x – 1
Sign Tables
  • They are used to visualize graph as it shows which quadrant the graph lies
  • Enter values of x which result in different parts of the fraction equaling zero
  • Leave columns between each value of x and place signs to indicate whether the value is +ve or -ve in each cell
  • Example
    • y = 3x² + 3x + 6/(x + 3)(x – 2)
Curve Sketching
  • When you sketch the curve, include the following;
    • y-intercept
    • x-intercepts
    • Stationary points (maxima, minima, inflections)
    • Vertical asymptote(s)
    • Horizontal or oblique asymptote(s)
    Examples
    1. The curve C has the equation y = x² – 4/x – 3
      1. Find the equations of the asymptotes
        • From the first formula:
          • deg(Q(x)) = deg(N(x)) – deg(D(x))
        • Write the equation as partial fractions:
          • y = x² – 4/x – 3 = Ax + B + C/x – 3
          • (Ax + B)(x – 3) + C = x² – 4
          • Ax² – 3Ax + Bx – 3B + C = x² – 4
        • Finding the coefficients using algebraic manipulation:
          • Ax² = x²
          • ∴ A = 1
          • -3Ax + Bx = 0x
          • ∴ -3A + B = 0
          • ∴ B = 3(1) = 3
          • -3B + C = -4
          • ∴ -3(3) + C = -4
          • ∴ C = -4 + 9 = 5
        • The final partial fraction can be expressed as:
          • y = x + 3 + 5/x – 3
        • Vertical asymptote:
          • x – 3 = 0
          • ∴ x = 3
        • Oblique asymptote:
          • as x increases, y ≈ x + 3
          • ∴ y = x + 3
      2. Draw a sketch of C and its asymptotes. Give the coordinates of the points of intersection of C with the coordinate axes
        • We know the asymptotes but need the intercepts and stationary points
        • y-intercept:
          • x = 0
          • ∴ y = 0 + 3 + 5/0 – 3 = 4/3
          • ∴ coordinates = (0, 4/3)
        • x-intercepts:
          • y = 0
          • ∴ x² – 4/x – 3 = 0
          • x² – 4 = 0
          • ∴ x = ±2
        • Sketch
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