Can someone please explain to me the steps of how to complete the square and then write the parabola y=x^2+6x+5 in the form (x-h)^2 = 4a(y-k).
Also from this how do I find the vertex, focus and the equation of the directrix.
Can someone please explain to me the steps of how to complete the square and then write the parabola y=x^2+6x+5 in the form (x-h)^2 = 4a(y-k).
Also from this how do I find the vertex, focus and the equation of the directrix.
$\displaystyle
\begin{aligned}y&=x^2+6x+5\\
y-5\bold{\color{red}+9}&=x^2+6x\bold{\color{red}+9}\ \y+4&=(x+3)^2\\4 \cdot \dfrac14 (y+4)&=(x+3)^2\end{aligned}$
Therefore:
$\displaystyle a=\dfrac14$ , h = -3, k = -4
(h, k) is the vertex.
a is the distance of the focus from the vertex.
a is the distance of the directrix from the vertex in the opposite direction from the focus.
For further information on the general "completing the square to find the vertex" process, you can review some online resources, following through their steps on the various worked examples.