# What are the vertex, focus and directrix of  y=-15+12x-2x^2 ?

Aug 8, 2018

$\left(3 , 3\right) , \left(3 , \frac{23}{8}\right) , y = \frac{25}{8}$

#### Explanation:

$\text{since the equation has an "x^2" term, this is a}$
$\text{vertically opening parabola}$

$\text{the equation of a vertically opening parabola is}$

•color(white)(x)(x-h)^2=4a(y-k)

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is the distance from the vertex to the focus and directrix}$

$\text{if "a>0" then opens upwards}$

$\text{if "a< 0" then opens downwards}$

$\text{to obtain this form "color(blue)"complete the square}$

$y = - 2 \left({x}^{2} - 6 x + \frac{15}{2}\right)$

$\textcolor{w h i t e}{y} = - 2 \left({x}^{2} + 2 \left(- 3\right) x + 9 - 9 + \frac{15}{2}\right)$

$\textcolor{w h i t e}{y} = - 2 {\left(x - 3\right)}^{2} + 3$

${\left(x - 3\right)}^{2} = - \frac{1}{2} \left(y - 3\right)$

$4 a = - \frac{1}{2} \Rightarrow a = - \frac{1}{8} \text{ parabola opens down}$

$\text{vertex } = \left(3 , 3\right)$

$\text{focus } = \left(h , a + k\right) = \left(3 , \frac{23}{8}\right)$

$\text{directrix is } y = - a + k = \frac{1}{8} + 3 = \frac{25}{8}$
graph{(y+2x^2-12x+15)(y-0.001x-25/8)=0 [-10, 10, -5, 5]}
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