# How do I use the vertex formula to determine the vertex of the graph for -3x^2+12x-9?

May 18, 2015

Vertex form: $f \left(x\right) = a {\left(x + \frac{b}{2 a}\right)}^{2} + f \left(- \frac{b}{2 a}\right)$

x of vertex:$x = \left(- \frac{b}{2 a}\right) = - \frac{4}{-} 2 = 2$

y of vertex: f(-2) = -12 + 24 - 9 = 3

Vertex form: $f \left(x\right) = - 3 {\left(x - 2\right)}^{2} + 3$

Check by developing:

$f \left(x\right) = - 3 \left({x}^{2} - 4 x + 4\right) + 3 = - 3 {x}^{2} + 12 x - 9$. OK

May 18, 2015

For $y = a {x}^{2} + b x + c$, the vertex formula tells us that the $x$-coordinate of the vertex is $- \frac{b}{2 a}$

For $y = - 3 {x}^{2} + 12 x - 9$, we have: $a = - 3$ and $b = 12$

Therefore, the vertex formula gives us $x$ coordinate is

$- \frac{12}{2 \left(- 3\right)} = - \frac{12}{- 6} = 1 \left(- 2\right) = 2$

We still need the $y$ coordinate of the vertex, so use the equation to find it:

$y = - 3 {\left(2\right)}^{2} + 12 \left(2\right) - 9 = - 12 + 24 - 9 = 3$

The vertex is $\left(2 , 3\right)$

graph{y=-3x^2+12x-9 [-10, 10, -5, 5]}