# What are the vertex, focus, and directrix of  y=-x^2+7x+5?

Oct 7, 2017

Vertex $\left(\frac{7}{2} , \frac{69}{4}\right)$
Focus $\left(\frac{7}{2} , 17\right)$
Directrix $y = \frac{35}{2}$

#### Explanation:

Given -

$y = - {x}^{2} + 7 x + 5$

This parabola opens down because it is in the form

${\left(x - h\right)}^{2} = - 4 a \left(y - k\right)$

Let us convert the given equation in this form

$- {x}^{2} + 7 x + 5 = y$
$- {x}^{2} + 7 x = y - 5$
${x}^{2} - 7 x = - y + 5$
${x}^{2} - 7 x + \frac{49}{4} = - y + 5 + \frac{49}{4}$
${\left(x - \frac{7}{2}\right)}^{2} = - y + \frac{69}{4}$
${\left(x - \frac{7}{2}\right)}^{2} = - 1 \left(y - \frac{69}{4}\right)$

${\left(x - \frac{7}{2}\right)}^{2} = - 4 \times \frac{1}{4} \left(y - \frac{69}{4}\right)$
$a = \frac{1}{4}$ Distance between focus and vertex and also distance between vertex and directix.

Vertex $\left(\frac{7}{2} , \frac{69}{4}\right)$
Focus $\left(\frac{7}{2} , 17\right)$
Directrix $y = \frac{35}{2}$