#### 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

- y =

**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}

- y =
- When x is very large, y =
^{3}/_{1}= 3 - Thus the horizontal asymptote at: y = 3

- y =

**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

- 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)}

- y =

**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**- The curve C has the equation y =
^{x² – 4}/_{x – 3}- 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

- y =
- 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

- From the first formula:
- 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

- Find the equations of the asymptotes