## Hyperbolas

Using a spreadsheet to find x and y values for the function and plotting the points, gives the graph of a hyperbola (plural is hyperbolas or hyperbolae).

The hyperbolas studied at this level are called rectangular hyperbolas because the asymptotes are vertical and horizontal. From the spreadsheet, the points can be plotted on a graph. Note that the value when x = 0 cannot be calculated and the line x = 0 is known as the vertical asymptote. The horizontal asymptote is the x axis, y = 0 As x gets nearer to being 0, the y value gets either very large or very small and the curve gets very close to the y axis but never touches it.

### The Basic Hyperbola

Equations of the type xy = c or have graphs which are hyperbolas.

 Example 1 Example 2 xy =5   Note: The bigger the value of c, the steeper the graph. Note: If c is negative the graph is in the 2nd and fourth quadrants.

### Transformation of the Basic Hyperbola

As with parabolas, circles and cubics some equations can be graphed using the transformation approach.

Example 1 . This is the hyperbola moved 2 units up the y-axis. Example 2 This is the basic hyperbola xy = 10 with the asymptotes moved forward 3 and down 4. ### The Hyperbola as an Algebraic Fraction (Preparation for Year 13) is a common form of an equation which has a hyperbola as its graph.

There are four main steps to sketching a graph of this type:

 Example: Sketch the graph of Step 1: Find the y-intercept Put x = 0 y-intercept is (0, -3) Step 2: Find the x-intercept Put y = 0 x-intercept is (-1.5, 0) Step 3: Find the vertical x-asymptote. (The denominator cannot be 0) Put the denominator = 0 x − 1 = 0x = 1 The x-asymptote is x = 1 Step 4: Find the horizontal y-asymptote The y asymptote is found by dividing out the fraction. A quick way to do this is to just divide the x terms. y-asymptote is The y-asymptote is y = 2 Sketch the graph 