Rukasha Maths

GCSE maths Beta

Straight-line graphs · Revision guide

Foundation · Algebra · 1 page in the PDF

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Learning objectives

  • Identify a gradient and vertical intercept.
  • Find the equation of a line from two points.

Key ideas

The gradient is the coefficient of the horizontal variable; the constant is the vertical intercept.

y=mx+cy=mx+c

Gradient is vertical change divided by horizontal change.

m=y2−y1x2−x1(x2≠x1)m=\frac{y_2-y_1}{x_2-x_1}\quad(x_2\ne x_1)

Find the vertical intercept by substituting a point into the line equation.

Distinct parallel non-vertical lines have equal gradients. A vertical line has undefined gradient.

Coordinates give the horizontal position first.

(x,y)(x,y)

Worked examples

Example 1

Find the line through the two given points.

(1,5), (3,9):m=9−53−1=2(1,5),\ (3,9):\qquad m=\frac{9-5}{3-1}=2

Substitute the first point.

5=2(1)+c⟹c=35=2(1)+c\quad\Longrightarrow\quad c=3

Check the second point.

2(3)+3=92(3)+3=9
y=2x+3y=2x+3

Example 2

Substitute a negative horizontal coordinate.

y=4x+1,x=−2:y=4(−2)+1=−7y=4x+1,\quad x=-2:\qquad y=4(-2)+1=-7
y=−7y=-7

Watch out

The constant is the vertical intercept, not the gradient. Keep brackets around negative values when substituting.