Graphs: use the form y = mx + c to identify perpendicular lines (Higher tier only)

The idea

For Higher tier, two straight lines are perpendicular if the product of their gradients is −1. In the form y = mx + c, if one line has gradient m, a perpendicular line has gradient −1/m (the negative reciprocal). For example, a line with gradient 2 is perpendicular to a line with gradient −1/2 because 2 × (−1/2) = −1. To identify perpendicular lines from equations, rearrange each into y = mx + c, read off the gradients, and check whether their product is −1. A horizontal line (gradient 0) is perpendicular to a vertical line, but vertical lines cannot be written in the form y = mx + c.

What earns the marks

• State that perpendicular lines have gradients whose product is −1.
• Find the negative reciprocal of a given gradient to obtain the perpendicular gradient.
• Rearrange a linear equation into the form y = mx + c to identify its gradient.
• Check whether two lines are perpendicular by multiplying their gradients and testing whether the result is −1.
• Recognise that a horizontal line (gradient 0) is perpendicular to a vertical line, which cannot be written as y = mx + c.

Common mistakes and how to put them right

• Thinking that perpendicular lines have the same gradient.

  Put it right by using the negative reciprocal relationship m₁ × m₂ = −1.

• Forgetting the negative sign when finding the perpendicular gradient.

  Put it right by changing the sign as well as flipping the fraction.

• Assuming that any two lines with different gradients are perpendicular.

  Put it right by checking that the product of the gradients is exactly −1.

• Trying to write a vertical line in the form y = mx + c.

  Put it right by recognising that vertical lines have equations of the form x = k and are perpendicular to horizontal lines.

Exam tips

• Rearrange equations into y = mx + c before comparing gradients.
• Write down the product of the two gradients to show clearly whether it equals −1.
• Remember that the negative reciprocal of a whole number m is −1/m, and of a fraction a/b is −b/a.