Question

How do you find the vertex and intercepts for `y = 3x^2 - 6x - 4`?

Answer 1

vertex: `(1,-7)`
y-intercept: `(-4)`
x-intercepts: `1+-sqrt(7/3)`

Explanation:

One way to find the vertex is to convert the given equation into vertex form:
`color(white)("XXX")y=3x^2-6x-4`

`color(white)("XXX")y=3(x^2-2x) -4`

`color(white)("XXX")y=3(x^2-2xcolor(green)(+1))color(green)(-3)-4`

`color(white)("XXX")y=3(x-1)^2+(-7)`
which is the vertex form for a parabola with vertex at `(1,-7)`

the y-intercept can be found by setting `x=0` in the original equation:
`color(white)("XXX")y=3(0)^2-6(0)-4=-4`

The x-intercepts are a bit more work, but can be found by setting `y=0` in any one of the equations and solving for `x`.
I find this easiest with the vertex form above:
`color(white)("XXX")0=3(x-1)^2-7`

`color(white)("XXX")3(x-1)^2=7`

`color(white)("XXX")(x-1)^2=7/3`

`color(white)("XXX")x-1=+-sqrt(7/3)`

`color(white)("XXX")x=1+-sqrt(7/3)~~2.53" or "-0.53`

A graph of the original equation might help show that these results are reasonable:
graph{3x^2-6x-4 [-4.13, 11.67, -7.17, 0.73]}