## Graph Basics

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:

$y = mx + c$

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 $y = 3x + 2$, 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 $y = 0$ and then solve to get $x = \frac{-2}{3}$.

If the $y$ in your equation has a number in front of it, for example, $2y = x +4$, divide by this number and the proceed as normal. In this example your equation would be $y = \frac{x}{2} + 2$.

#### Useful Fact

If two lines are parallel, their lines have equal gradients. So for the lines $y = mx + c$ and $y = nx + d$, if $m = n$, then these two lines are parallel, it doesn’t matter what the $c$ and $d$ are.

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 $y = ax^2 + bx + c$, if $a$ is greater than zero, then y is a positive quadratic with a ‘u’ shaped graph, seen on the left hand side. If $a$ 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, $c$ is the y intercept.

### Constructing Graphs

Straight Line Graphs

These are actually quite simple to draw. Let’s take the graph $y = 5x -4$.

You already have one point in the form of your y intercept $4$, we now need to find out where the graph crosses the x axis. It does this when $y = 0$, so setting $y = 0$ we get $0 = 5x - 4$, hence $x = \frac{4}{5} = 0.8$.

We have now got two points, $(0,-4)$ and $(0.8,0)$. If we plot these on our axes and draw a line through them, we have our graph!

These are a little bit more complicated but the sames rules apply. For examplw, draw the graph of $y = x^2 - x - 6$.

The number in front of the $x^2$ is positive, so we know the curve will be ‘u’ shaped. We also know that the y intercept is $-15$.

Next we need to find where the graph intercepts the x axis. To do this we set $y = 0$ and factorise to get $(x - 3)(x + 2) = 0$, so the graph intercepts the x axis at $x = -2, 3$.

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 $(x-1)^2$, you single value of $x$ 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).