The equation of a curve is y = (x + 3)^2 + 5. Find the coordinates of the turning point.

We are asked to find the coordinates of the turning point of a line, and we should first remind ourselves of what this means. A turning point on a line is either a maximum or minimum point, or a point of inflection. (These may be easily represented on a graph). They are all points of zero gradient. When we are given the equation of a line and are asked to find an equation for the gradient, what do we do? We differentiate it!
So if y = (x + 3)^2 + 5, then dy/dx = 2 x (x + 3). (Here we used basic differentiation rules which can be revised if the student requires).
Because we are finding the points with zero gradient, we must put dy/dx = 0 which implies that 2 x (x + 3) = 0, which in turn implies that x = -3.
So we know our x-coordinate, but we must substitute this in to our equation of a line so that we can find the y-coordinate. y = (x + 3)^2 + 5, so at our turning point y = ((-3) + 3)^2 +5 which implies that y = 5.
So we have found that x = -3 and y = 5, and therefore the coordinates of the turning point are (-3, 5).