Gradients of Straight Line Graphs Worksheets | Questions and Revision

# Gradients of Straight Line Graphs Worksheets, Questions and Revision

Level 4 Level 5

## What you need to know

The gradientof a line is a measure of how steepit 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.

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

Example: Find the gradient of the line graph shown

$\textcolor{red}{\text{Change in } y}$ $= 3 -(-1) = 4$

$\textcolor{blue}{\text{Change in } x}$$= 2 - 0 = 2$

$\text{Gradient } = \dfrac{\textcolor{red}{4}}{\textcolor{blue}{2}} = 2$

A Positive gradient – $y$ value Increases as the $x$ value increases.

A negative gradient – $y$ value decreases as the $x$ value increases.

## Example: Gradient from 2 co-ordinates

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

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

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

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

### Example Questions

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 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.

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:

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

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