A curve has equation y = x^3 - 48x. The point A on the curve has x coordinate -4. The point B on the curve has x coordinate - 4 + h. Show that that the gradient of the line AB is h^2 - 12h.

First, let's recall the gradient formula:grad = change in y / change in x = y2 -y1 / x2 - x1So let's label point A as x1, and point B as x2, and find the corresponding y1 and y2 using the equation of the curve.x1 = -4, y1 = (-4)^3 - 48 (-4) = 128x2 = -4 + h, y2 = (-4 + h) ^ 3 - 48 (-4 + h) = h^3 - 12 h^2 + 48 h - 64 - 48 h + 192 (by expanding the brackets) = h^3 - 12 h^2 + 128 (by simplifying the expression)so gradient AB = (h^3 - 12 h^2 + 128 - 128) / (-4 + h - -4) = (h^3 - 12 h^2) / h = h^2 - 12h as required.