WASSCE SHS
The gradient of a line through two points is the change in y divided by the change in x. One point and the gradient fix the line.
For points (x₁, y₁) and (x₂, y₂), the gradient m is (y₂ − y₁) divided by (x₂ − x₁). The order must be the same on the top and the bottom. Swapping only the y values changes the sign. A horizontal line has gradient 0. A vertical line has an undefined gradient, because the change in x is zero and you cannot divide by zero. Parallel lines have equal gradients. Perpendicular lines have gradients whose product is −1, provided neither line is vertical.
The equation of a line of gradient m through (x₁, y₁) is y − y₁ = m(x − x₁). Rearrange into the form you were asked for. If you are asked for y = mx + c, expand and collect the constant. Check by substituting both given points. Each should satisfy the equation. If the second point fails, the gradient's sign is the first place to look.
A gradient of 2 from the points (1, 3) and (5, 11) is (11 − 3)/(5 − 1). Show that fraction. The equation mark needs a point used, not only the gradient. y = 2x + 1 should be checked at both points if you have time. Perpendicular gradients multiply to −1. The negative reciprocal of 2 is −1/2, not −2.
These are practice questions in the style of the paper, with a model answer and a mark scheme. They are not a promise of what your paper will ask.
5 marks
A line passes through (1, 3) and (5, 11).
(a) Find the gradient. (2 marks)
(b) Find the equation of the line in the form y = mx + c. (3 marks)
(a) m = (11 − 3)/(5 − 1) = 8/4 = 2.
(b) y − 3 = 2(x − 1) y = 2x − 2 + 3 y = 2x + 1.
2 marks
Find the gradient of a line perpendicular to y = 2x + 1. (2 marks)
The given gradient is 2. The perpendicular gradient is −1/2, because 2 × (−1/2) = −1.
1 mark
Explain why a vertical line has no gradient in this formula. (1 mark)
The change in x is zero, and division by zero is undefined.
More questions on this topic are inside GLearn, marked as you answer, with the reason written out. The first ones need no account.
Practise Elective Mathematics freeChange in y divided by change in x, in the same order. From (1, 3) to (5, 11) the gradient is 8/4, which is 2. The line is y = 2x + 1.
The negative reciprocal, so that the product of the gradients is −1. A line of gradient 2 is perpendicular to a line of gradient −1/2. This does not apply to a vertical line, which has undefined gradient. All WASSCE Elective Mathematics past questions