Linear Functions and Graphs

In Grade 7 you learned that every straight line has the equation $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept. Give you the equation, and you can picture the line.

This year you work the other way. Given a graph, or even just two points, you will find the equation yourself.

Why does this matter? In the real world, a graph often comes first — you measure data, plot it, and see a straight line. Finding its equation lets you make predictions and compare relationships.

Keep your Grade 7 toolkit close — gradient, intercept and $y = mx + c$ are still the heart of everything here.

When a straight line is already drawn, finding its equation is a two-part job: read off the gradient $m$, then read off the $y$-intercept $c$.

For the gradient, draw a step triangle between two clear points on the line. Count the rise, count the run, and divide. If the line goes up 4 for every 2 across, $m = 4 \div 2 = 2$.

For the intercept, simply look where the line crosses the $y$-axis. If it crosses at 1, then $c = 1$.

Putting the two together, the equation is $y = 2x + 1$.

A couple of careful points:

• If the line slopes down, the gradient is negative.
• Always pick step-triangle points where the line passes exactly through grid corners — that keeps your reading accurate.

Once you have $m$ and $c$, just slot them into $y = mx + c$.

Sometimes there is no drawn line — just two points. That is still enough to find the equation.

Suppose a line passes through $(1, 5)$ and $(3, 11)$.

Step one — find the gradient. The gradient is the change in $y$ divided by the change in $x$:

$m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3$

Step two — find the intercept. You know $m = 3$, so the equation so far is $y = 3x + c$. Substitute one of the points — say $(1, 5)$ — and solve for $c$:

$5 = 3 \times 1 + c \quad\Rightarrow\quad c = 2$

So the full equation is $y = 3x + 2$. Always check with the other point: $3 \times 3 + 2 = 11$ — correct.

This two-step method works for any pair of points on a straight line.

Two lines are parallel when they run in the same direction and never meet. There is a beautifully simple test for this in algebra.

Parallel lines have exactly the same gradient $m$.

So $y = 2x + 1$, $y = 2x + 5$ and $y = 2x - 3$ are all parallel — every one has gradient 2. They cross the $y$-axis at different points, but they slope in exactly the same way.

This makes some questions quick to answer. If a line is parallel to $y = 4x - 7$ and passes through $(2, 5)$, you already know its gradient is 4. Then find $c$ as before: $5 = 4 \times 2 + c$ gives $c = -3$, so the line is $y = 4x - 3$.

The opposite is also worth knowing: lines with different gradients are not parallel — they will cross at exactly one point somewhere.

Straight-line graphs describe the real world. On a real-life graph, the gradient and intercept always carry meaning.

A few examples of what the parts mean:

• On a cost graph, the $y$-intercept is a fixed charge and the gradient is the price per unit.
• On a distance-time graph, the gradient is the speed.
• On a graph of a savings plan, the intercept is the starting balance and the gradient is the amount saved each week.

To compare two real-life lines, look at both numbers. A steeper line has a larger rate; a higher intercept means a larger starting amount. Two phone tariffs, for instance, can be compared instantly from their graphs.

To interpret a graph fully, describe the story in words: the starting amount, the rate of change, and what happens where lines cross.

Being able to find the equation of any line opens the door to a great deal of maths ahead:

• Simultaneous equations are solved where two lines cross — finding their equations comes first.
• Direct proportion graphs are straight lines through the origin, a special case of $y = mx + c$ with $c = 0$.
• Science is full of straight-line relationships, and finding the equation lets you state the rule a graph shows.
• In everyday life, finding equations lets you compare tariffs, predict savings and model costs.

Later you will meet curved graphs, but the language of gradient and intercept you sharpen here stays useful throughout.

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