Question

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

Answer 1

The intercepts are `(0,0)` and `(4,0)`. The vertex is `(2, -4)`.

Explanation:

The intercepts are when one of the two variables is equal to zero.

If `x=0` we have `y=0`

When `y=0` we have

`0=x^2-4x` we can say that `x!=0` (because this solution we already found) and divide both side by `x`

`0/x=(x^2-4x)/x`
`0=x-4` and then we have also the solution `x=4`.

We have intercepts for the two points `(0,0)` and `(4,0)`.
The parabola is symmetric with respect to the vertex, so the `x` of the vertex has to be in the middle of the two `x` of the intercepts.
The two `x` are `0` and `4` then the `x` of the vertex is `2`.
The `y` can be obtained substituting the `x` into the equation:

`y=2^2-4*2=4-8=-4`.
The vertex is then in the point `(2, -4)`.

graph{x^2-4x [-10, 10, -5, 5]}