How do you find the turning points of a curve described by the equation y(x)?

A turning point on a curve will be a point where the curve is flat/horizontal i.e. where it has a gradient of zero. We know that to find the equation for the gradient of a curve we have to differentiate it with respect to x. To find the turning point(s) we then have to set the equation for the gradient equal to zero and solve the equation to find the value(s) of x which satisfies the equation. Once we know this value we can substitute it back into the original equation for the curve to find the correct value of y- We will have found the x and y coordinates of the turning point! Heres a straightforward example: Find the turning point of the curve y = x² - 8x + 19.