Answer:
[tex]\large\boxed{y=\dfrac{1}{4}x^2-x-4}[/tex]
Step-by-step explanation:
The equation of a parabola in vertex form:
[tex]y=a(x-h)^2+k[/tex]
(h, k) - vertex
The focus is
[tex]\left(h,\ k+\dfrac{1}{4a}\right)[/tex]
We have the vertex (2, -5) and the focus (2, -4).
Calculate the value of a using [tex]k+\dfrac{1}{4a}[/tex]
k = -5
[tex]-5+\dfrac{1}{4a}=-4[/tex] add 5 to both sides
[tex]\dfrac{1}{4a}=1[/tex] multiply both sides by 4
[tex]4\!\!\!\!\diagup^1\cdot\dfrac{1}{4\!\!\!\!\diagup_1a}=4[/tex]
[tex]\dfrac{1}{a}=4\to a=\dfrac{1}{4}[/tex]
Substitute
[tex]a=\dfrac{1}{4},\ h=2,\ k=-5[/tex]
to the vertex form of an equation of a parabola:
[tex]y=\dfrac{1}{4}(x-2)^2-5[/tex]
The standard form:
[tex]y=ax^2+bx+c[/tex]
Convert using
[tex](a-b)^2=a^2-2ab+b^2[/tex]
[tex]y=\dfrac{1}{4}(x^2-2(x)(2)+2^2)-5\\\\y=\dfrac{1}{4}(x^2-4x+4)-5[/tex]
use the distributive property: a(b+c)=ab+ac
[tex]y=\left(\dfrac{1}{4}\right)(x^2)+\left(\dfrac{1}{4}\right)(-4x)+\left(\dfrac{1}{4}\right)(4)-5\\\\y=\dfrac{1}{4}x^2-x+1-5\\\\y=\dfrac{1}{4}x^2-x-4[/tex]
