G4

Question: Find the equation of the straight line graph shown to the right

Solution: The equation of a straight line takes the form $y=mx+c$, where we need to find $m$, the gradient, and $c$, the $y$-intercept.

We can’t see the $y$-intercept on the graph, so we’ll have to work out the gradient first and get to $c$ in a moment.

To find the gradient of a straight line, we draw a right-angled triangle underneath the line, and divide the change in $y$ (the height) by the change in $x$ (the width). This looks like:

The height of this triangle is 3, and the width is 1. The only other thing to consider is that the line is sloping downwards, so the gradient will be negative. So, we get

$\text{gradient }=-\dfrac{3}{1}=-3$

Now, to find $c$, we have to firstly pick a pair of coordinates that the line passes through – here we’ll choose (-1, 0). Then, we have to substitute this $x$ value and $y$ value into the equation $y=mx+c$, along with the $m$ that we just found.

Doing this, we get the equation

$0=(-3)\times(-1)+c$

Simplifying and rearranging this equation, we get

$0=3+c$

$c=-3$

So, we’ve found that $c=-3$ and the gradient $m=-3$, so the equation of the line is

$y=-3x-3$