Question

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

Answer 1

Vertex: `(2,0)`
y-intercept: `4`
x-intercept: `2`

Explanation:

The general vertex form for a parabola is:
`color(white)("XXX")y=(x-color(red)(a))^2+color(blue)(b)`
for a parabola with vertex at `(color(red)(a),color(blue)(b))`

`y=x^2-4x+4`

`color(white)("X")=(x-color(red)(2))^2+color(blue)(0)`
is therefore a parabola with vertex at `(color(red)(2),color(blue)(0))`

`bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXX"))`

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

Given `y=x^2-4x+4`
when `x=0`
`color(white)("XXX")y=4`

`bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXX"))`

The x intercept is the value(s) of `x` when `y=0`

Given `y=x^2-4x-4`
when `y=0`
`color(white)("XXX")0=x^2-4x+4`

`color(white)("XXX")x^2-4x+4=0`

`color(white)("XXX")(x-2)^2=0`

`color(white)("XXX")(x-2)=0`

`color(white)("XXX")x=2`