Question

What is the standard form of a quadratic function with a vertex of (-2,-3) and passing through the point (-4,-1)?

Standard form is `y=a(x-p)^2+q`

Answer 1

Standard form: `y=1/2x^2+2x-1.`
Vertex form: `y=1/2(x+2)^2-3.`

Explanation:

Standard form is `y=ax^2+bx+c`. Vertex form is `y=a(x-p)^2+q,` where `(p,q)` are the coordinates of the vertex.

As such, it is easy to find vertex form for a parabola with the given information. That form needs values for `a,p,` and `q.` We already have `p` and `q`, so to find `a`, all we do is plug in what we know to the vertex form equation:

`color(white)(–)y=a("  "x" "-"   "p"   ")^2+"   "q`
`–1=a(–4-(–2))^2+(–3)`
`–1=a(–4+2)^2-3`
`color(white)(–)2=a(–2)^2`
`color(white)(–)2=4a`

`"  "1/2=a`

So vertex form for this parabola is `y=1/2(x+2)^2-3.`

To find standard form, we expand the right hand side:

`y=1/2(x+2)^2-3`

`color(white)y=1/2(x^2+4x+4)-3`

`color(white)y=1/2x^2+2x+2-3`

`color(white)y=1/2x^2+2x-1`

So standard form for this parabola is `y=1/2x^2+2x-1.`