y = mx + c

Any equation that can be rearranged into the form \(y = mx + c\), will have a straight line graph. \(m\) is the gradient, or steepness of the graph, and \(c\) is the \(y\)-intercept, or where the line crosses the \(y\)-axis.

Finding c

The graphs of \(y = 2x + 1\) and \(y = 2x - 2\) are shown below.

The graph of \(y = 2x + 1\) crosses the \(y\)-axis at (0, 1). The graph of \(y = 2x - 2\) crosses the \(y\)-axis at (0, -2). The constant term in the equation (the + 1 or ‒ 2) shows the point where the graph crosses the \(y\)-axis.

This is known as the \(y\)-intercept and is represented by the letter \(c\) in \(y = mx + c\).

Finding m

The graphs \(y = 2x\) and \(y = 4x\) are shown below:

The number in front of the \(x\) is the of the graph.

As you move along a line from left to right, you might go up, you might go down or you might not change at all.

Gradient = \(\frac{\text{change up}}{\text{change right}}\) or \(\frac{\text{change in y}}{\text{change in x}}\)

Gradients can be:

positive – going up
negative – going down
zero - no change (a flat line)

Steep lines will have high gradients like 5 or -8 and equations like \(y = 5x – 4\) or \(y = -8x + 1\).

Fairly flat lines will have low gradients like \(\frac{1}{2}\) or \(- \frac{3}{4}\) and equations like \(y = \frac{1}{2}x + 1\) or \(y = - \frac{3}{4} x + 2\).

Horizontal lines have a gradient of 0 and equations like \(y = 2\).

To work out a gradient, use the scales of the axes and find how many units you go up or down for each unit you move right.

To work out the equation of a line from a graph, find the gradient and the \(y\)-intercept.

Example 1

Work out the equation of this graph.

Gradient = \( \frac{\text{change up}}{\text{change right}}\)

The gradient is the same all along the line, so it doesn’t really matter where you start or finish, but it is generally a good idea to use two points on the line that are far apart.

Using (0, 3) and (4, 7), as we move along the line from left to right, we move 4 units up (from 3 to 7) and 4 units to the right (from 0 to 4). So the gradient \(= \frac{4}{4} = 1 \).

The \(y\)-intercept is 3 because the line crosses the \(y\)-axis at (0, 3).

So the equation of the line in the form \(y = mx + c\) is \(y = 1x + 3\) or just \(y = x + 3\).

Example 2

Work out the equation of this graph.

A straight line graph that plots the equation y = -2x + 1. A red line passes through minus 3 and 7 on the Y axis and 4 and minus 7 on the X axis.

Gradient = \(\frac{\text{change up}}{\text{change right}}\)

Using (0, 1) and (4, −7), as we move along the line from left to right, we move 8 units down (from 1 to −7). We also move 4 units right (from 0 to 4).

So the gradient = \(\frac{-8}{4} = -2\).

The \(y\)-intercept is 1 because the line crosses the \(y\)-axis at (0, 1).

So the equation of the line is \(y = -2x + 1\).

Example 3

Work out the equation of this graph.

A straight line graph that plots the equation y = 1 over 3 times – 2. A red line passes through minus 6 and minus 4 on the Y axis and 6 and zero on the X axis.

Gradient = \(\frac{\text{change up}}{\text{change right}}\)

For example, using (0, −2) and (6, 0), as we move along the line from left to right, we move 2 units up (from −2 to 0) and 6 units to the right (from 0 to 6).

So the gradient = \(\frac{2}{6} = \frac{1}{3}\).

The \(y\)-intercept is −2 because the line crosses the \(y\)-axis at (0, −2).

So the equation of the line is \(y = \frac{1}{3}x - 2\).

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Straight line graphs - Eduqasy = mx + c

y = mx + c

Finding c

Finding m

Example 1

Example 2

Example 3

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