Straight-Line Graphs — y = mx + c

Every straight-line graph can be written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). Knowing these two values is enough to draw or describe any straight line.

The Equation of a Straight Line

$y = mx + c$

Letter	Meaning
$m$	Gradient — how steep the line is (rise ÷ run)
$c$	y-intercept — where the line cuts the y-axis

Identify m and c

$y = 3x + 2$

Gradient: $m = 3$ (line goes up 3 for every 1 across)
y-intercept: $c = 2$ (crosses y-axis at (0, 2))

Drawing a Straight-Line Graph

Plot the y-intercept $(0, c)$
Use the gradient: from that point, go across 1 and up $m$ (or down if $m$ is negative)
Plot the second point and draw a line through both

Draw y = 2x − 1

y-intercept: $(0, -1)$
Gradient 2 → go right 1, up 2 → second point at $(1, 1)$
Draw the line

Reading the Equation from a Graph

Read off the y-intercept → $c$
Choose two clear points; calculate gradient = (change in $y$ ) ÷ (change in $x$ )

Rearranging to Slope-Intercept Form

Sometimes equations are given in other forms. Rearrange to isolate $y$ :

Rearrange 2y − 4x = 6

$2y = 4x + 6$ $y = 2x + 3$

Gradient = 2, y-intercept = 3

Tip

Parallel lines have the same gradient but different y-intercepts. E.g. $y = 3x + 1$ and $y = 3x − 4$ are parallel.

Real world: mobile phone plan

A plan costs £10/month plus 5p per minute. The cost equation is $y = 0.05x + 10$ .

For 200 minutes: $y = 0.05 \times 200 + 10 = £20$ . The gradient (0.05) is the per-minute charge; the y-intercept (10) is the fixed fee.

Real world: temperature conversion

Converting Celsius to Fahrenheit uses $F = 1.8C + 32$ — a straight line with gradient 1.8 and y-intercept 32.

At $C = 100$ : $F = 1.8 \times 100 + 32 = 212°F$ (the boiling point of water).

Real world: taxi fare

A taxi charges £2.50 base fare plus £1.40 per km. The fare equation is $y = 1.4x + 2.5$ . A 10 km ride costs $1.4 \times 10 + 2.5 = £16.50$ . Comparing two taxi companies means comparing their gradients and intercepts.

Practice Problems

State the gradient and y-intercept of y = −2x + 5.

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m = −2 (line slopes downward), c = 5 (crosses y-axis at (0, 5)).

A line passes through (0, 3) and (2, 7). Write its equation.

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m = (7−3)/(2−0) = 2. c = 3. Equation: y = 2x + 3.

Rearrange 3y + 9x = 12 into the form y = mx + c.

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3y = −9x + 12 → y = −3x + 4. m = −3, c = 4.