Graphs are a great way to represent data in a straightforward and simple way. Whether you’re showing off the number of visitors to your Facebook page, or an accountant portraying the latest stock forecasts, graphs are essential.

### Equations of a straight line

One of the simplest graphs is that of a straight line. These are written in the form:

In this form, **m** is the **gradient** of the line, (the steepness), and **c** is the **y intercept**, (where the line cuts the y axis).

In the above graph for , we can clearly see that the line has a gradient of 3 and y intercept 2. If we want to find the x intercept, we set and then solve to get .

If the in your equation has a number in front of it, for example, , divide by this number and the proceed as normal. In this example your equation would be .

#### Useful Fact

If two lines are parallel, their lines have equal gradients. So for the lines and , if , then these two lines are parallel, it doesn’t matter what the and are.

### Quadratic Graphs

As you can see from the picture above, Quadratic Graphs are curves with a turning point or ‘stationary’ point. Consider a quadratic in the form , if is greater than zero, then y is a **positive quadratic** with a ‘u’ shaped graph, seen on the left hand side. If is less than zero, then y will be a **negative quadratic** with a ‘n’ shaped graph, seen here on the right.

As with the straight line examples, is the y intercept.

### Constructing Graphs

**Straight Line Graphs**

These are actually quite simple to draw. Let’s take the graph .

You already have one point in the form of your y intercept , we now need to find out where the graph crosses the x axis. It does this when , so setting we get , hence .

We have now got two points, and . If we plot these on our axes and draw a line through them, we have our graph!

**Quadratic Graphs**

These are a little bit more complicated but the sames rules apply. For examplw, draw the graph of .

The number in front of the is positive, so we know the curve will be ‘u’ shaped. We also know that the y intercept is .

Next we need to find where the graph intercepts the x axis. To do this we set and factorise to get , so the graph intercepts the x axis at .

Using these 3 points and the shape of the graph, we can the draw the following:

When drawing quadratic curves your graph doesn’t have to be perfect, but spotting the shape of the graph and the intersecting points is essential.

**Important Note**

If, when finding the values of the x intercept, you end up with a repeated root, for example , you single value of will be 1. Then the graph just ‘touches’ the axis at this point. The graph below is an example of this.

**Solving Equations Using Intersecting Graphs**

If we have the graphs of two equations and you are asked to solve them simultaneously, the point(s) where the two graphs meet are the simultaneous value(s).