# Functions, Conics & Asymptotes

## Domain & Range

• The domain of a function is the set of all valid/possible input values
• The x axis
• The range of a function is the set of all possible output values
• The y axis

## Conics

Equation of a circle with radius and centre

Equation of an ellipse with centre , major axis length and minor axis length :

Equation of a Hyperbola with vertex :

The asymptotes of this hyperbola are at:

## Asymptotes

There are 3 kinds of asymptotes:

• Vertical
• Horizontal
• Oblique (have slope)

For a function :

• Vertical asymptotes lie where and
• Horizontal asymptotes
• If the degree of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x-axis
• If the degree of the numerator is bigger than the degree of the denominator, there is no horizontal asymptote.
• If the degrees of the numerator and denominator are the same, the horizontal asymptote equals the leading coefficient of the numerator divided by the leading coefficient of the denominator
• Oblique asymptotes
• A rational function will approach an oblique asymptote if the degree of the numerator is one order higher than the order of the denominator
• To find
• Divide by
• Take the limit as

Example: find the asymptotes of :

• Vertical asymptotes:
• Where the denominator is 0

• Horizontal asymptotes:
• There are none, as degree of the numerator is bigger than the degree of the denominator
• Oblique asymptotes:
• Divide the top by the bottom using polynomial long division
• Find the limit

As , , giving as an asymptote.