What is the the vertex of x = (y -3)^2 - 9 ?

Nov 25, 2015

The vertex coordinates are (3, -9).

Explanation:

Let us consider that the variables were inverted on purpose. That way, y is the horizontal axis and x is the vertical one.

First of all, solve the Mathematical Identitiy:
${\left(y - 3\right)}^{2} = \left(y - 3\right) \cdot \left(y - 3\right) = {y}^{2} - 3 y - 3 y + 9$
Then simplify the function:
$x = {y}^{2} - 3 y - 3 y - 9 + 9 = {y}^{2} - 6 y$

From this point on, there are many ways to find the vertex. I prefer the one that doesn't use formulas. Every quadratic formula takes the shape of a parabola, and every parabola has a symmetry axis. That means points that have the same height have the same distance from the center. Therefore, let's calculate the roots:

$y \left(y - 6\right) = 0$
$y ' = 0$
$y ' ' \to y - 6 = 0$
$y ' ' = 6$

Find the point that is between the roots: $\frac{0 + 6}{2} = 3$. Therefore, $y v = 3$. Now, to find the x value corresponding, just solve the function for 3:
$x \left(3\right) = {\left(3\right)}^{2} - 6 \cdot \left(3\right) = 9 - 18 = - 9$.

Therefore, the axis is located at (3, -9).
graph{(x-3)^2-9 [-2, 8, -10, 10]}