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 – 1} – ³/_{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