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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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