## What you need to know

The gradient of a line is a measure of how steep it is. If the gradient is small, the slope of the line will be very gradual, but if the gradient is big, the line will be quite steep. You are required to know how to calculate the gradient from two possible circumstances:

• You are given the line drawn on a graph;
• You are given two coordinates and told that a line passes through both of them.

These situations may seem distinct, but they aren’t really. They both come down to the same idea:

$\text{Gradient } = \dfrac{\text{change in }y}{\text{change in }x}$

Additionally, the gradient is also a rate of change, specifically a rate of change of $y$ with respect $x$ – this is why we divide the change in $y$ by the change in $x$ to calculate it. What this all means is this: the gradient of a line tells us how much the $y$ value changes when the $x$ increases by 1. For example, if you have a line with gradient 3, then every time the $x$ value increases by 1, the $y$ value increases by 3.

Let’s see about calculating gradient given a line that’s been drawn for us.

Example: Calculate the gradient of the line drawn below.

To do this, you want to pick 2 points on the graph that the line passes through. It’s best, if you can, to pick two points where the coordinates are easy to read off.

Here, we picked $(2, 1)$ and $(4, 5)$, as seen on the graph on the right. Once you’ve done this, draw the right-angled triangle as pictured with dotted lines. Then, the change in $x$ is the width of the base of that triangle, whilst the change in $y$ is the height.

Therefore, we get

$\text{gradient } = \dfrac{4}{2} = 2$.

You should be wary of the possibility of a negative gradient.

We just saw the positive gradient case. In the negative gradient case, the $y$ value decreases as the $x$ value increases. Fear not, you can still use the triangle method seen above for working out gradient, but if the line is sloped downwards then you’ll need to put a minus sign in front of your answer.

As mentioned at the beginning, you might be asked to calculate the gradient given two coordinates that a line passes through. Let’s see an example.

Example: Work out the gradient of the straight line that passes through $(2, 3)$ and $(-10, 6)$.

To find the change in $x$ and change in $y$ here, we must pick one of the points and subtract its $x$ and $y$ coordinates from the other point’s $x$ and $y$ coordinates respectively. Subtracting the first from the second, we get

$\text{gradient } = \dfrac{-10 -2}{6 - 3} = \dfrac{-12}{3} = -4$.

In general, if we have two coordinates $(x_1, y_1)$ and $(x_2, y_2)$ then the gradient of the line that passes through them is

$\text{gradient } = \dfrac{\text{change in }y}{\text{change in }x} = \dfrac{y_2 - y_1}{x_2 - x_1}$

Equivalently, you could calculate it by doing

$\dfrac{y_1 - y_2}{x_1 - x_2}$.

What matters is that you’re subtracting the coordinates of one point from the coordinates of the other, i.e. the order of your subtraction is the same on top and bottom. In doing this, the answer will tell you itself whether the gradient is positive or negative, and you don’t have to worry.

## Example Questions

#### 1) Calculate the gradient of the line drawn below.

We must find two points that the line passes through and draw a right-angled triangle underneath, so we can identify the change in $x$ to be the base and the change in $y$ to be the height. This looks like

Now, given that this is a downwards slope, it must have a negative gradient. So, we get

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

Note: you could’ve used a different triangle at different points on the line – this is fine, as long as you got the correct answer of -3.

#### 2) Find the gradient of the line that passes through $(-8, -1)$ and $(2, -6)$.

To find the gradient, we’ll subtract the values of second coordinate from those of the first, and divide the difference in the $y$ values by the difference in the $x$ values:

$\text{gradient } = \dfrac{-1 - (-6)}{-8 - 2} = \dfrac{5}{-10} = -\dfrac{1}{2}$

## Gradients of Straight Line Graphs Revision and Worksheets

Straight Line Graphs
Level 1-3
Straight Line Graphs 2
Level 1-3