Question

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

Answer 1

vertex at `(2,-1)`
y-intercept: `3`
x-intercepts: `3` and `1`

Explanation:

To find the vertex rewrite the given equation `y=x^2-4x+3`
into vertex form: `y=m(x-a)^2+b` with vertex at `(a,b)`

Complete the square
`color(white)("XXX")y=x^2-4xcolor(blue)(+2^2)+3color(blue)(-4)`
Rewrite as a squared binomial and simplified constant (in vertex form)
`color(white)("XXX")y=1(x-2)^2+(-1)`
with vertex at `(2,-1)`

The y-intercept is the value of `y` when `x=0`

For `y=x^2-4x+3` when `x=0`
`color(white)("XXX")y=9^2-4xx(0)+3 = 3`

The x-intercepts are the values when `y=0`

Since `y=x^2-4x+3`
can be factored as `y=(x-3)(x-1)`
when `y=0`
`color(white)("XXX")x=3` or `x=1`