Linear functions and straight-line graphs (y = mx + c)

Functions grow in many different ways, and each way describes how something develops in real life. It could be how the money in a bank account grows with a certain interest rate, or how the price of a taxi ride grows for every kilometre you drive. Money in a bank account and a taxi ride are described by different types of function, but when we talk about linear functions, we mean things that grow linearly. That is, the graph of the function is a straight line. A lot of situations out in the real world are described by what we call a linear function.

The equation of a linear function

Every type of function has its own equation, a form the function has to fit before it counts as that type. A linear function has the equation

$$y = mx + c$$

This says that the function value, $y$, is equal to the matching $x$-value multiplied by a number $m$, plus a constant $c$. You will often see the same line written as $f(x) = ax + b$ in other books. The letters are different, the line is exactly the same. In the UK the gradient is called $m$ and the constant is called $c$, so that is what we use here.

What m and c mean

In the equation there are two values we don't know to begin with, $m$ and $c$.

$m$ 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.

As you might guess, that means the bigger $m$ is, the steeper the line. The gradient $m$ tells us something about how much the function slopes. If the line went 4 up the y-axis every time it goes 1 along the x-axis, $m$ would be 4. We can also have gradients that are negative. Then the graph slopes downwards, and for every 1 we go along the x-axis, the line goes down instead of up.

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

If the function hits the y-axis at 2, $c$ is 2. We call $c$ the y-intercept. The guide on gradient and y-intercept goes through both of them with more examples and pictures.

Why c is where the line crosses the y-axis

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 exactly what $c$ describes.

Setting up a linear function

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, and the function crosses the y-axis at 3. Then we can 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.

Linear functions in real life

When we work with linear functions in real life, a meaning like "crossing with the y-axis" can seem a bit odd. A taxi ride helps. Say a taxi costs £2 per km, and there is a start fee of £4. Every kilometre adds £2 to the price, and that is exactly what $m$ is: how much we go up the y-axis when we go 1 along the x-axis. The start fee is what we pay before we have driven anything at all, so that is where the price starts on the y-axis, our $c$. The function for the price becomes

$$y = 2x + 4$$

That is also why GCSE calls the gradient a rate of change: here it is the £ per km. The whole example, with the graph, is in the guide on gradient and y-intercept.

Finding the equation from two points

Often we need the gradient of a linear function without being told it. Typically we have been given two points on the line instead. For that there is a formula, which funnily enough is called the two-point formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Once we have $m$, we find $c$ by putting one of the points into the equation. The full walkthrough is in the guide on finding the equation of a straight line from two points.

Straight lines through data

Sometimes we don't have a neat line at all, just a lot of points from measurements, and we want the straight line that describes those points as well as possible. That is called linear regression, or at GCSE a line of best fit. It is not a different topic: it just finds the $m$ and $c$ of the line that fits best. See the guide on scatter graphs and the line of best fit.

Common misunderstandings

• "m is just a number you multiply by." It is that, but it is also a picture. $m$ is how far the line goes up (or down) every time you move 1 to the right. If you can read that off a graph, you can read off $m$.
• "c is where the word problem starts, so it changes as you go." No. $c$ is where the line crosses the y-axis, the value when $x$ is 0. In the taxi example $c$ is the £4 start fee, and you pay it no matter how far you drive.
• "£6 for 3 km means the gradient is 6." No. The gradient is how much we go up when we go exactly 1 along the x-axis, not 3. £6 for 3 km is £2 per km, so $m$ is 2. Always rewrite the information so it says how much $y$ grows for 1 step in $x$.
• "A line of best fit is a different kind of maths." It isn't. Linear regression just finds the $m$ and $c$ of the straight line that fits a set of points best.

Related guides

• Gradient and y-intercept: what m and c mean
• Finding the equation of a straight line from two points
• Scatter graphs and the line of best fit
• A linear function is a type of function, and direct proportion is a linear function with $c = 0$.