At the turning points of an equation the slope of y is zero. In other words the differential of the equation must be zero. So for your example:
So we set this equal to zero to get:
or x=0
I don't know if this thread is right for the forum..
but I have a maths assignment where it's asking to state the turning point of certain equations but i don't know how to find it.
e.g
y=x^2
how to i find the turning point of that parabola?
Since the question is posted in the Pre-calculus subforum I have to assume that a non-calculus approach is required.
@OP: You're probably expected to know the turning point form of a parabola and how to get it. Certainly you should know that has a turning point at (h, k).
There are three approaches to finding the turning point of a parabola.
Method 1) Due to the symmetry of the parabola, the turning point lies halfway between the x-intercepts.
If there is only one x-intercept, then the x-intercept IS the turning point.
So in your case, for , the x-intercept is found by letting .
So .
Since there is only one x-intercept, that is the x-coordinate of the turning point.
To find the y-coordinate, substitute it into your equation.
.
So the turning point is .
For another example, look at the parabola given by the equation .
To find the x-intercept, let .
So
or
or .
To find the x-coordinate of the turning point, average the x-intercepts.
So .
To find the y-coordinate, substitute this value into the equation.
.
So the turning point is .
Method 2)
Method 1 is ok to use if x-intercepts exist. However, it is inappropriate to use this technique if x-intercepts do not exist.
Therefore, most people prefer to use the "Turning Point Form" of the parabola.
For any parabola written in the form
its turning point is at .
This form of the parabola is found by completing the square.
With your example, , we can rewrite it as
.
So its turning point is .
With the other example, , we complete the square to put it into turning point form.
.
So the turning point is .
Method 3) is to use Differential Calculus, but since this is in the Pre-Calculus forum, I doubt you'd be using this method.