Linear Functions and Graphs

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.

• Every straight line has the equation $y = mx + c$.
• This year you find the equation from a graph or from points.
• Two points are enough to pin down a line.
• Finding equations lets you predict and compare.

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 $y = mx + c$.

• Read the gradient with a step triangle: rise divided by run.
• Read the intercept where the line crosses the $y$-axis.
• A downward-sloping line has a negative gradient.
• Slot $m$ and $c$ into $y = mx + c$.

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.

• Find the gradient: change in $y$ divided by change in $x$.
• Write $y = mx + c$ with the gradient filled in.
• Substitute one point and solve for $c$.
• Check the equation with the other point.

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.

• Parallel lines have exactly the same gradient.
• Their intercepts can be different.
• A line parallel to a known line copies its gradient straight away.
• Lines with different gradients always cross once.

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

• On a cost graph the intercept is a fixed charge.
• On a distance-time graph the gradient is the speed.
• A steeper line means a larger rate of change.
• Compare two graphs using both gradient and intercept.

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.

• Crossing lines solve pairs of simultaneous equations.
• Direct proportion is $y = mx + c$ with intercept zero.
• Science states relationships as straight-line equations.
• Equations of lines model tariffs, savings and costs.