Straight Line Graphs

• Every straight line can be written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where it crosses the y-axis).
• You can find the equation of a line from a graph, from a gradient + one point, or by rearranging a messy equation into y = mx + c form.
• Horizontal lines are y = c. Vertical lines are x = k. (These are the two "special case" lines students forget!)
• You can draw a straight line from a table of values, from y = mx + c directly, or by finding the x- and y-intercepts of ax + by = c.
• Parallel lines have the same gradient — always. Nothing else matters.
• Perpendicular lines have gradients that are negative reciprocals of each other: m₁ × m₂ = −1.
• A perpendicular bisector cuts a line segment exactly in half, at a right angle — you need the midpoint AND the perpendicular gradient to find it.

Think of y = mx + c as a recipe for a straight line. Every straight line — no matter how it's drawn or where it sits on the page — can be described using just two numbers: m and c.

Here's the intuition: c tells you where to start on the y-axis. m tells you how much to climb (or fall) every time you move 1 step to the right. That's genuinely the whole idea — everything else in this chapter is just applying that.

Finding the equation of a line from a graph

If you're handed a graph and asked for its equation, follow this sequence every time:

1. Find the gradient. Pick any two clear points the line passes through. Draw an imaginary right-angled triangle between them and calculate: gradient = rise ÷ run.
2. Check the direction. If the line slopes downhill (left to right), the gradient is negative — don't forget the minus sign!
3. Read off the y-intercept. Look at exactly where the line crosses the y-axis.
4. Substitute m and c into y = mx + c. Done.

What if the y-intercept isn't shown?

Sometimes the graph is zoomed in, or you're just given a gradient and a point — no visible y-intercept. No problem: treat c as an unknown, substitute in the point you DO know, and solve for it algebraically.