Linear Functions and Graphs

In Grade 6 you learned to read coordinates, plot points from a table of values, and join them into a straight line. A linear function always gives a straight-line graph.

This year you go to the heart of that line. Every straight line can be captured in a single equation, and two numbers control everything about it: how steep it is and where it crosses the $y$-axis.

Why does this matter? Once a line has an equation, you no longer need to plot lots of points. You can sketch it instantly, compare two lines without drawing them, and predict values far beyond your table.

Keep your Grade 6 plotting skills sharp — this guide builds straight on top of them.

The gradient of a line measures how steep it is. The idea is simple: pick any two points on the line and ask "how far up for each step across?".

$\text{gradient} = \frac{\text{change in } y}{\text{change in } x}$

Draw a step triangle between two points: count how far the line rises, count how far it runs across, then divide. If a line rises 6 for every 2 across, the gradient is $6 \div 2 = 3$.

The sign tells you the direction:

• A line sloping up from left to right has a positive gradient.
• A line sloping down has a negative gradient.
• A flat line has a gradient of 0.

The size matters too: a gradient of 5 is steeper than a gradient of 2. Whatever two points you pick on the same straight line, the gradient always comes out the same.

The second number that pins down a line is its $y$-intercept — the point where the line crosses the $y$-axis.

At that crossing point, $x$ is always 0. So the $y$-intercept is simply the value of $y$ when $x = 0$.

The $y$-intercept can be positive, negative or zero. A line through the origin has a $y$-intercept of 0.

In a real-world graph, the $y$-intercept often has a meaning all of its own. On a graph of taxi cost against distance, the $y$-intercept is the fixed charge before you have travelled anywhere. On a graph of a tank emptying, it is the amount of water you started with.

Together, the gradient and the $y$-intercept describe a line completely — and the next section ties them into one neat equation.

Here is the key idea of this whole topic. Every straight line has the equation

$y = mx + c$

where $m$ is the gradient and $c$ is the $y$-intercept. Read the equation and you can picture the line; picture the line and you can write the equation.

For example, $y = 3x + 2$ is a line with gradient 3 and a $y$-intercept of 2. The line $y = -2x + 5$ slopes downward (negative gradient) and crosses the $y$-axis at 5.

This equation gives you a fast way to plot a line:

1. Mark the $y$-intercept $c$ on the $y$-axis — that is one point.
2. Use the gradient $m$ as a step: across 1, up $m$, to reach the next point.
3. Repeat the step, then join the points with a ruler.

No full table needed. The equation $y = mx + c$ turns drawing a line into a quick, two-number job.

Once lines are written as $y = mx + c$, you can compare them without drawing a single point.

Some quick comparisons:

• $y = 5x + 1$ is steeper than $y = 2x + 1$, because 5 is further from zero than 2.
• $y = 3x + 4$ and $y = 3x - 1$ have the same gradient, so they are parallel — they never meet.
• $y = -4x + 2$ slopes downward; $y = 4x + 2$ slopes upward. They cross the $y$-axis at the same point.

To sketch a line from its equation, you do not need precise points — just get the two features right: start at the correct $y$-intercept, and slope it the right way with roughly the right steepness.

This is the real payoff of $y = mx + c$: lines become things you can read, compare and picture at a glance.

Straight-line graphs are not just abstract — they describe the real world. On these graphs, the gradient and intercept carry meaning.

A few examples of what the parts mean:

• On a distance-time graph, the gradient is the speed — a steeper line means faster travel.
• On a cost graph, the $y$-intercept is a fixed charge and the gradient is the price per item.
• A flat section means no change — a stationary object, or a cost that is not rising.

To read a value, do exactly what you practised in Grade 6: go up from the $x$-axis to the line, then across to the $y$-axis.

To interpret a graph, describe the story in words: where it is steep, where it is flat, where it rises or falls, and what the crossing points mean. A graph is a picture of something happening — your job is to put that picture into sentences.

Mastering $y = mx + c$ now opens the door to a great deal of maths ahead:

• Solving equations can be done with graphs — the point where two lines cross is the solution that satisfies both.
• Sequences and graphs meet: a linear sequence plots as evenly spaced points, and the common difference becomes the gradient.
• Science is full of straight-line relationships — speed, current, and rates of change all appear as gradients.
• In everyday life, straight-line graphs model phone tariffs, currency conversion, fuel use and savings plans.

Later you will meet curved graphs too, but the language of gradient and intercept you build here stays useful — you will talk about how steep a curve is at a point using exactly these ideas.

If graph work feels hard later, it is usually the gradient calculation that needs a refresh. Come back to this guide whenever you need it — there is no prize for rushing.