There are some other types of graph that you should be familiar with:
Quadratic Graphs and Quadratic Equations
You may be asked to use a graph to solve a quadratic. For example solve the equation 0 = x2 + 2x – 3 using the graph y = x2 + 2x – 3 (shown above).
This is easy! Just look to see where y = 0 on the graph i.e. where the graph crosses the x axis.
So here you can see the answers are -3 and 1:
X Cubed Graphs
These have a kink in them. How big the kink is depends on the other terms in the equation.
So here is y = x3
and here is y = x3 + 2x2 + 1:
and here is y = –x3 + 2x2 + 1 (notice it slants downwards from left to right):
So if asked to draw a cubed graph just plot enough points and join them by a smooth curve.
The graphs are in the form y = kx
If k is greater than 1 then the graph slopes up from left to right.
If k is less than 1 or is negative the graph slopes down from left to right.
They always go through the point (0,1) and are always above the x axis.
So here is the graph for y = 2x:and this is what it would look like if 2 was to the power MINUS x (see the surds page to realise this is actually 2 to the power a half):Reciprocal Graphs
These graphs occur when you have the basic formula y = A/x
This could be written xy = A
They only ever exist in two quadrants and they don’t touch.
Which quadrants they are in depends if they are positive or negative.
The graph for y = 1/x looks like this:and y = -1/x looks like:
The equation for a circle with centre (0,0) and radius r is:
- x2 + y2 = r2
So x2 + y2 = 9 is a circle with centre (0,0) and a radius of 3:
Sine And Cos Waves
These waves wiggle up and down between one and minus one on the y axis. They repeat every 360 degrees.
Between 0 and 360 degrees the sine wave forms a nice wave. The cos wave forms a bucket between these values. Below you can see the sine wave in red and the cos wave in blue: