Question

How do you write the vertex form equation of the parabola `y = -2(x+4)(x-2)`?

Answer 1

Please see the explanation.

Explanation:

Given: `y = -2(x+4)(x-2)`

Multiply the factors:

`y = -2(x^2+4x-2x -8)`

`y = -2(x^2+2x -8)`

`y = -2x^2-4x +16" [1]"`

Please observe that equation [1] is in the standard form `y = ax^2+bx+c` where `a = -2, b = -4 and c = 16`

The vertex form of a parabola of this type is:

`y=a(x-h)^2+k" [2]"`

The "a" in the standard form and the "a" in the vertex form are the same attribute of the parabola, therefore, substitute -2 for "a" into equation [2]:

`y=-2(x-h)^2+k" [3]"`

We know that `h = -b/(2a)`:

`h = -(-4)/(2(-2)`

`h = -1`

Substitute -1 for h into equation [3]:

`y=-2(x--1)^2+k" [4]"`

Evaluate equation [1] at `x = 0`

`y = 16`

Evaluate equation [4] and x = 0 and y = 16, then solve for k:

`16=-2(1)^2+k`

`k = 18`

The vertex form is:

`y=-2(x--1)^2+18" [5]"`