Question

How do you find the vertex and intercepts for `y = x^2 - 9`?

Answer 1

Vertex: `(0,-9)`
`y"-intercept: " y=-9`
`x"-intercepts: " x=-3" and " x=+3`

Explanation:

The general vertex form is
`color(white)("XXX")y=color(green)m(x-color(red)a)^2+color(blue)bcolor(white)("xxx")`with vertex at `(color(red)a,color(blue)b)`

The given equation `y=x^2-9` can be written explicitly in vertex form as
`color(white)("XXX")y=color(green)1(x-color(red)0)^2+color(blue)(""(-9))color(white)("xxx")`with vertex at `(color(red)0,color(blue)(""(-9)))`

The `y`-intercept is simply the value of `y` when `x=0`.
Substituting `0` for `x` in the given equation:
`color(white)("XXX")y=0^2-9rArry=-9`

Similarly, the `x`-intercepts are the values of `x` which solve the equation when `y=0`.
Substituting `0` for `y` in the given equation:
`color(white)("XXX")0=x^2-9`
`color(white)("XXX")rarr x^2=9`
`color(white)("XXX")rarr x=+-3`