Question

Math smarties needed!

Answer 1

1.
`y = (x + 2)(x + 3)`

2.
`y = -x(x + 4)`

Explanation:

We can find the factored form of the graphed quadratic functions by:

1) Expressing each one in vertex form

2) Expanding and refactoring to rewrite in factored form

1. We know that vertex form is:

`y = a(x-h)^2 + k`,

where
`a` determines the parabola's direction and width (
`a=1` being the same as a standard parabola), and
`(h,k)` is the parabola's vertex.

The vertex of this parabola is (-3, -4).

↓ plugging these values into the vertex form equation

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

↓ simplifying

`y = (x + 3)^2 - 4`

Now, we can expand and refactor this equation into factored form.

↓ expanding the squared term

`y = (x^2 + 6x + 9) - 4`

↓ simplifying

`y = x^2 + 6x + 5`

↓ factoring

`\boxed{y = (x + 2)(x + 3)}`

2. We can see that the vertex is at (-2, 4).

↓ plugging into the vertex form equation

`y = -1(x - (-2))^2 + 4`

Note that the parabola opens downward, so
`a = -1`.

↓ simplifying

`y = -1(x +2)^2 + 4`

↓ expanding the squared term

`y = -1(x^2 +4x + 4) + 4`

↓ incorporating the outer +4 into the distribution of -1

`y = -1(x^2 + 4x + 4 - 4)`

↓ simplifying

`y = -1(x^2 + 4x)`

↓ factoring

`y = -1(x)(x + 4)`

↓ simplifying

`\boxed{y = -x(x + 4)}`