# How do I use the vertex formula to determine the vertex of the graph for f(x) = −x^2 + 18x − 75?

May 1, 2015

The vertex formula has the form
$y = m {\left(x - a\right)}^{2} + b$ for constants $m , a , b$
with a vertex at $\left(a , b\right)$

Converting $f \left(x\right) = - {x}^{2} + 18 x - 75$
into the vertex formula form is a matter of "completing the square"

$\left(f x\right) = \left(- 1\right) \left({x}^{2} - 18 x + {9}^{2}\right) - 75 + {9}^{2}$

$= \left(- 1\right) {\left(x - 9\right)}^{2} + 6$

The vertex is at $\left(9 , 6\right)$

May 1, 2015

The vertex formula says that the graph of $y = a {x}^{2} + b x + c$ has its vertex at the point where $x = - \frac{b}{2 a}$.

In this equation, we have $a = - 1$ and $b = 18$.

The formula tells us that the $x$-coordinate of the vertex is:

$x = - \frac{18}{2 \left(- 1\right)} = - \frac{18}{-} 2 = - \left(- 9\right) = 9$

To find the $y$-coordinate, use the equation and $x = 9$:

$y = - {\left(9\right)}^{2} + 18 \left(9\right) - 75$

$y = - 81 + 162 - 75 = 162 - 156$

$y = 6$

The vertex is $\left(9 , 6\right)$