How to find the minimum point on a quadratic curve?

Or highest point for that matter?

E.X: I have the equation y=x^2-8x+23

Take the derivative and equate to 0.

\(\displaystyle y^{ \prime } = 2x - 8 = 0 \)

Min happens at

\(\displaystyle x = 4 \)

If we don't use derivatives (which I suspect you are not supposed to) we complete the square.

\(\displaystyle y = x^2-8x+23 = (x^2 - 8x + 16 - 16) + 23 = (x-4)^2 + 7 \)

From here we can find the max/min. Since this parabola is facing upwards, we know there in a minimum.

This is known as "vertex form" and the min/max points happen at the vertex. In this case,

the vertex is \(\displaystyle (4,7) \) which comes from the negative of the number inside the bracket with the x, and the number that is added.

In general

\(\displaystyle y = (x-p)^2 + k \)

has the vertex

\(\displaystyle (p,k) \)