Gradient and y-intercept: what m and c mean

In the equation of a straight line, $y = mx + c$, there are two values we don't know to begin with: $m$ and $c$. This guide shows what each of them means, how to read them off a graph, and how to find them in a real situation like a taxi fare. The usual hurdle is the second half: a meaning like "crossing with the y-axis" can feel odd until you have seen it as a start fee.

When do I use this?

Every time you meet a linear function. If you are given a graph, you read $m$ and $c$ off it. If you are given a situation in words, you pick $m$ and $c$ out of the information and write the equation yourself. And whenever a question asks what the gradient means in context, the answer is always the same shape: how much $y$ grows for every 1 step in $x$.

The gradient m

$m$ is the gradient of the line. The gradient of a straight line is defined as how far we go up the y-axis when we go 1 along the x-axis. In the picture you can see the definition of what the gradient is.

If the function went 4 up the y-axis when it goes 1 along the x-axis, $m$ would be 4. As you might guess, that means the bigger $m$ is, the steeper the straight line. The gradient $m$ tells us something about how much the function slopes.

Negative gradients

We can easily have gradients that are negative. If we had a gradient of $-2$, the graph would look roughly like this:

Notice how the graph now slopes downwards, because $m$ is negative. In this example, every time we go 1 along the x-axis, we go 2 down the y-axis.

The y-intercept c

$c$ is rather simpler. It describes where the function crosses the y-axis.

If the function hits the y-axis at 2, then $c$ is 2.

How do we know that $c$ describes the crossing with the y-axis? All the points on the y-axis have one thing in common: their $x$-values are all 0. That is what it means to be on the y-axis. So if we want to find the place where the function is on the y-axis, we just put 0 in $x$'s place in the equation of the line:

$$y = mx + c$$

$$y = m \times 0 + c$$

We can see that the term with $m$ disappears completely, because it is multiplied by 0, and so we only have $c$ left:

$$y = c$$

We have now shown that the crossing point between the y-axis and our straight line is what $c$ describes.

Setting up a linear function from a description

Let's take an example with both $m$ and $c$. Say a function goes 5 up the y-axis when it goes 1 along the x-axis. The function also crosses the y-axis at 3. We can now set up the function:

$$y = 5x + 3$$

$m$ is 5 because the function goes 5 up the y-axis when it goes 1 along the x-axis. $c$ is 3 because the function crosses the y-axis at 3.

m and c in real life: the taxi

When we work with linear functions in real life, a meaning like "crossing with the y-axis" can seem odd, so let's look at an example that helps with the understanding.

As we know, $m$ shows how much we go up the y-axis when we go 1 along the x-axis. Say we order a taxi, and we would like to set up a function for the price, so we can work out exactly what it will cost for every km we drive. We are told that it costs £2 per km, and that there is also a start fee of £4.

We can start by seeing which variables we need on the x-axis and the y-axis. Here it looks like the price depends on how far we drive, so we put km on the x-axis and the price on the y-axis. That is what makes the most sense.

We know a linear function comes in the form $y = mx + c$, so we just need to find $m$ and $c$. We are told that when we have driven 1 km, we pay £2. That is exactly what $m$ is: how much we go up the y-axis when we go 1 along the x-axis. So we have a gradient of 2.

But we also see that there is a start fee of £4, so when we get into the taxi, we start at £4. That matches our $c$, the crossing with the y-axis. We pay £4 even though we have driven 0 km. So we have to remember that the function starts at 4 on the y-axis, and then grows by £2 per km.

We have now made the equation of the straight line:

$$y = mx + c$$

$$y = 2x + 4$$

At GCSE the gradient is also called a rate of change, and here you can see why: it is simply the £ per km. For 3 km the price is $2 \times 3 + 4 = 10$, so £10.

We have to remember that $m$ describes the growth on the y-axis when we go 1 along the x-axis. If we were told £6 for 3 km, the gradient is not 6, because then we are told that we go 6 up when we go 3 along. We have to rewrite the information so it shows how much we go up the y-axis when we go only 1 along, not 2, or 5, or something else. £6 for 3 km is $\frac{6}{3} = 2$ pounds per km, so $m$ is still 2.

When we solve problems where we have to set up a function or find $m$ and $c$, we should always look for these two types of information: how much we go up the y-axis when we go 1 along the x-axis, and any start value.

Common mistakes

• "m is just the number in front of x." It is that, but it is also something you can see. $m$ is how far the line rises (or falls) every time you move 1 to the right. Use the picture: a small step of 1 along, and read how far up you went.
• "c is the value at the start of the story, so it is part of the growth." No. $c$ is where the line crosses the y-axis, the value when $x = 0$. The £4 start fee is paid whether you drive 1 km or 20 km. It never grows.
• "£6 for 3 km means m = 6." The gradient is the growth for exactly 1 step along the x-axis. Rewrite the information first: £6 for 3 km is £2 per km, so $m = 2$.

Related

• Back to the pillar: linear functions and straight-line graphs.
• When you are given two points instead of the gradient: finding the equation of a straight line from two points.
• When you are given lots of points from data: scatter graphs and the line of best fit.