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O Level · International Mathematics 0607

Straight Line Graphs

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Summary — What This Chapter Covers

  • 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.

1. The Equation y = mx + c

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.

The Formula y = mx + c
m = gradient (how steep the line is, and whether it goes uphill or downhill)
c = y-intercept (the height at which the line crosses the y-axis, i.e. where x = 0)

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.

Worked Example

y = 5x + 2 is a straight line with gradient 5 and y-intercept 2. It starts at height 2 on the y-axis, and climbs 5 units for every 1 unit you move right — a steep uphill line.

y = 3 − 4x is a straight line with gradient −4 and y-intercept 3. It starts at height 3, and since the gradient is negative, it falls 4 units for every 1 unit you move right — a steep downhill line.

Sign check Positive gradient = uphill (line rises left to right). Negative gradient = downhill (line falls left to right). A gradient of zero means a flat, horizontal line.

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.
Worked Example — reading a graph

A line passes through (2, 4) and (10, 0), and crosses the y-axis at 5.

Rise = 4, Run = 8, so gradient = 4/8 = 1/2. But the line slopes downhill, so m = −1/2.

y-intercept: c = 5

Equation: y = −½x + 5

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.

Worked Example

A line has gradient 6 and passes through (2, 15). Find its equation.

Write y = 6x + c

Substitute x = 2, y = 15:   15 = 6(2) + c → 15 = 12 + c

Solve: c = 3

Equation: y = 6x + 3

Don't Forget A horizontal line has equation y = c (e.g. y = 4) — no x term at all, because the height never changes. A vertical line has equation x = k (e.g. x = −2) — it has no gradient in the y = mx + c sense (its gradient is technically undefined/infinite). These trip up almost everyone at least once!
📝 Practice Question 1 A straight line has a gradient of −2 and passes through the point (3, 1). Find the equation of the line.
📝 Practice Question 2 Write down the equations of: (a) a horizontal line passing through (3, −5), and (b) a vertical line passing through (7, 2).

2. Drawing Straight Line Graphs

There are three ways to draw a straight line, and which one you use just depends on what information you're given.

Method A — From a table of values (no equation given)

Take each (x, y) pair as coordinates, plot the points, then join them with a ruler. Every point should lie exactly on your line — if one doesn't, you've made an arithmetic slip somewhere.

Method B — Using y = mx + c directly

Build your own table of values by choosing 2–3 x-values spread across the given axis range, substituting each into the equation to find y, plotting those points, and joining them.

Worked Example

Plot y = 2x + 1 for x from 0 to 10. Choose spread-out x-values: 0, 5, 10.

x0510
y11121

Plot (0,1), (5,11), (10,21) and connect with a ruler.

Method C — The "staircase" method (no table needed)

This is the fastest method once you're confident. Assuming the equation is in y = mx + c form:

  1. Plot a single starting point at the y-intercept, c.
  2. From there, for every 1 unit you move right, move up m units (down if m is negative).
  3. If m is a fraction a/b, move a units up for every b units right — this avoids messy fraction-of-a-square jumps.
  4. Repeat this stepping pattern to build a sequence of points, then join them up.
Example: y = 2x + 1, starting at (0, 1) y 6 | * 5 | * 4 | * 3 | * 2 | * 1 | * <-- start here (y-intercept) 0 +------------------- x 0 1 2 3 Each step: +1 right, +2 up (gradient = 2)
Watch out Be careful when the axes have different scales — 1 unit on the graph might not equal 1 grid square! Always check the axis labels before using the staircase method.

What if the equation isn't in y = mx + c form?

Equations are often given messily, like 3x + 5y = 30. You have two options:

Option 1 — Rearrange it into y = mx + c form:

Worked Example

Rearrange 3x + 5y = 30:

Subtract 3x from both sides: 5y = −3x + 30

Divide everything by 5: y = −⅗x + 6

Now you can see gradient = −⅗ and y-intercept = 6.

Option 2 — Use the intercepts method (often faster for equations in ax + by = c form):

Worked Example

For 3x + 5y = 30:

Set x = 0:   5y = 30 → y = 6  → point (0, 6)

Set y = 0:   3x = 30 → x = 10 → point (10, 0)

Plot these two points and join with a ruler — done, no rearranging needed!

Examiner Tip Always plot at least 3 points, not just 2. If your third point doesn't sit on the line joining the first two, you know immediately that one of your calculations is wrong — before you lose marks.
📝 Practice Question 3 Rearrange 2x + 4y = 20 into the form y = mx + c, and state the gradient and y-intercept.

3. Parallel Lines

Parallel lines are straight lines that never meet — they run side-by-side forever, always the same distance apart. The reason they never meet is simple: they climb at exactly the same rate.

The Rule If a line is parallel to y = mx + c, its equation is y = mx + d
Same gradient (m stays identical). Only the y-intercept (d) can be different.

y = 2x + 1 and y = 2x + 5 ARE parallel (same gradient, 2).
y = 2x + 1 and y = 3x + 1 are NOT parallel (different gradients).

Finding the equation of a parallel line

The method is identical to finding any line's equation from a gradient and a point — the only twist is that you're copying the gradient from the given line rather than being told it directly.

Worked Example

Find the equation of the line parallel to y = 3x + 7, passing through (2, 1).

Same gradient, so write: y = 3x + d

Substitute x = 2, y = 1:   1 = 3(2) + d → 1 = 6 + d

Solve: d = −5

Equation: y = 3x − 5

📝 Practice Question 4 Find the equation of the line parallel to y = 4x − 2 that passes through the point (1, 9).

4. Perpendicular Lines

Perpendicular lines meet at a perfect right angle (90°) — think of the corner of a piece of paper, or a "+" sign. Unlike parallel lines, their gradients are related in a very specific (and slightly less obvious) way.

The Rule m₁ × m₂ = −1
The two gradients are "negative reciprocals" of one another. To find a perpendicular gradient, flip the fraction upside down AND change the sign.

Here's a fast way to think about "negative reciprocal": flip it, then flip the sign.

Original gradientPerpendicular gradient
1−1
1/3−3
−2/33/2
4−1/4
Alternative form m₂ = −1 / m₁

Checking if two lines are perpendicular

Get both equations into y = mx + c form (rearranging first if necessary), then check whether their gradients are negative reciprocals.

y = ⅓x + 10 and y = −3x − 18 ARE perpendicular (⅓ × −3 = −1 ✓)
y = ⅐x + 16 and y = 7x − 8 are NOT perpendicular (⅐ × 7 = 1, not −1)

Finding the equation of a perpendicular line

Same overall structure as before — but now the first step is calculating the flipped-and-negated gradient before you can even begin.

Worked Example

The line L has equation y − 2x + 2 = 0. Find the equation of the line perpendicular to L, passing through (2, −3).

Step 1 — Rearrange L: y − 2x + 2 = 0 → y = 2x − 2. Gradient of L is 2.

Step 2 — Flip and negate: perpendicular gradient = −1/2

Step 3 — Substitute the point into y = −½x + c:   −3 = −½(2) + c → −3 = −1 + c → c = −2

Equation: y = −½x − 2

Common Mistake Students often forget to flip the SIGN as well as the fraction. A gradient of 2 becomes −½, not ½. Missing the negative sign is one of the most common lost marks in this whole topic.

Perpendicular Bisectors

A perpendicular bisector is a special perpendicular line: it doesn't just meet a line segment at 90°, it also cuts that segment exactly in half. This means you need two pieces of information the perpendicular line must satisfy: (1) it passes through the midpoint of the segment, and (2) its gradient is the negative reciprocal of the segment's gradient.

Midpoint Formula (needed here) Midpoint = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
Worked Example

Find the equation of the perpendicular bisector of the segment joining (4, −6) and (8, 6).

Step 1 — Midpoint: ((4+8)/2, (−6+6)/2) = (6, 0)

Step 2 — Gradient of the segment: (−6−6)/(4−8) = −12/−4 = 3

Step 3 — Perpendicular gradient: −1/3

Step 4 — Substitute the midpoint into y = −⅓x + c:   0 = −⅓(6) + c → 0 = −2 + c → c = 2

Equation: y = −⅓x + 2

📝 Practice Question 5 Find the gradient of the line perpendicular to a line with gradient −4.
📝 Practice Question 6 Find the equation of the perpendicular bisector of the line segment joining (0, 2) and (6, 8).

What to Memorise

y = mx + c The general equation of any straight line. m = gradient, c = y-intercept.
Gradient formula gradient = rise ÷ run. Positive = uphill, negative = downhill.
Horizontal line y = c (a fixed height, no x term).
Vertical line x = k (a fixed left-right position, no y term).
Parallel lines Same gradient (m), different y-intercept. y = mx + c and y = mx + d.
Perpendicular lines m₁ × m₂ = −1. Flip the gradient's fraction and change its sign.
Midpoint formula ( (x₁+x₂)/2, (y₁+y₂)/2 ) — needed for perpendicular bisectors.
Intercepts method For ax + by = c: set x = 0 to find y-intercept, set y = 0 to find x-intercept.

Concepts Checklist

Exam Tips & Common Mistakes

Forgetting the negative sign on downhill lines. If a line slopes downhill left-to-right, its gradient MUST be negative. This is one of the single most common lost marks in this topic — always double-check the direction before writing your final gradient.
Only flipping the fraction, not the sign, for perpendicular gradients. A gradient of 5 becomes −1/5, not 1/5. Always do both steps: flip AND negate.
Mixing up horizontal and vertical line equations. Remember: y = c is horizontal (like the word "flat"), x = k is vertical. If you're unsure, sketch a tiny diagram in the margin — takes 5 seconds and saves the mark.
Leaving 5y instead of y when rearranging. Examiners want the equation fully solved for a single y — always divide through completely so you have exactly "y = ..." on one side.
Forgetting to find the midpoint for perpendicular bisector questions. A perpendicular bisector needs BOTH the correct gradient AND the midpoint — students often calculate a correct perpendicular gradient but then substitute the wrong point (like one of the original endpoints instead of the midpoint).
Not checking the requested answer format. Some questions ask for the answer in the form ax + by + c = 0 with integer coefficients — if you leave your answer as y = −½x − 2, you may lose a mark for not fully following the instruction. Always multiply through to clear fractions when asked.
Plotting only 2 points when drawing a graph. Always plot a third point as a check — if it doesn't lie on the line formed by the first two, you've made an error somewhere and can fix it before losing marks.
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