Question

How do you find the vertex and the intercepts for `y= -2(x+3)(x-1)`?

Answer 1

Vertex: Find AoS, sub in as `x` to find y-component.
X-intercepts: it's in factored form.
Y-intercept: sub in `x` as `0` and solve for `y`.

Explanation:

So intercepts are when the parabola touches the axes.

It is when one variable 0, thus, we can sub in `0` and solve for the other variable.

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X-intercepts

That is the zeros AKA roots AKA solutions. The equation is in factored form, giving us the zeros without having to calculate it. Thus, the x-intercepts are `(-3, 0)` and `(1, 0)`.

Y-intercept

We can find the `y"-intercept"` in two ways: changing the equation into standard form, OR sub `x` as `0`. We'll do the easier method, subbing in.

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

`y=-2(0+3)(0-1)`

`y=-2(3)(-1)`

`y=6`

Therefore, the `y"-intercept"` is `(0, 6)`.

Vertex

To find vertex in factored form, the easiest method is to find the axis of symmetry, and sub that in as `x` and solve for `y`.

The axis of symmetry can be calculated given the formula: `x=(r+s)/2`.
=> `r` and `s` are the zeros.
=> `x` is the axis of symmetry AKA the `x"-component"` in the vertex.

Finding AoS

`x=(r+s)/2`

`x=(-3+)/2`

`x=-2/2`

`x=-1`

Subbing in AoS to find y-component of vertex

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

`y=-2(-1+3)(-1-1)`

`y=-2(2)(-2)`

`y=8`

Therefore, the vertex is `(-1, 8)`.

Let's check our work by graphing it.

graph{-2(x+3)(x-1) [-10, 10, -0.24, 9.76]}

As you can see, the vertex and intercepts are correct.

Hope this helps :)