Question

How do you find the vertex and intercepts for `y=(-x-1)(x+7)`?

Answer 1

`y`-intercept: `y=-7`
`x`intercepts: `x=-1` and `x=-7`
vertex at: `(-4,9)`

Explanation:

Given
`color(white)("XXX")y=(-x-1)(x+7)`

The y-intercept is the value of `y` when `x=0` (that is, on the Y-axis, since for all values on the Y-axis, `x=0`)
Substituting `0` for `x` in the given equation:
`color(white)("XXX")y=(-0-1)(0+7)=-7`
The y-intercept is at `y=-7`

The x-intercepts are the values of `x` for which `y=0` (that is on the X-axis)
`color(white)("XXX")0=(-x-1)(x+7)`

#color(white)("XXX"){:((-x-1)=0," or ",(x+7)=0), (rarr x=-1,,rarrx=-7):}#
The x-intercepts are at `x=-1` and `x=-7`

The vertex can be determined in a couple ways:
1. by converting the given equation into vertex form (ask if you need to see this version)
or
2. by noting that the axis of symmetry will cross the X-axis a t he mid point between the 2 x-intercepts;
namely at `x=((-1)+(-7))/2==-4`
Substituting `x=-4` into the given equation gives the `y` coordinate of the vertex
`color(white)("XXX")y=(-(-4)-1)((-4)+7)=3 * 3 =9`
So the vertex is at `(-4,9)`