A curve C has equation y = (2 - x)(1 + x) + 3 . A line passes through the point (2, 3) and the point on C with x-coordinate 2 + h . Find the gradient of the line, giving your answer in its simplest form.
First we find the y coordinate which is a function of x:
x = 2+ h so y = (2 - 2 - h)(1 + 2 + h) + 3 = -h2 - 3h + 3
Now for the gradient, the line passes through points (2,3) and (2 + h, -h2 - 3h + 3)
dx = 2 - 2 - h = -h dy = 3 + h2 + 3h - 3 = h2 +3h
The gradient dy/dx = -(h + 3)
