Question

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

Answer 1

`color(blue)("Vertex " -> (x,y)-> (2, -2)`;
`ycolor(white)(.)_("intercept")=2`

`xcolor(white)(.)_("intercpts") ->` I have taken you to the point where all you have to do is the final arithmetic

Explanation:

Given: `g(x)=x^2-4x+2`

The quick way for some of it!
There is no coefficient in front of `x^2` so we can use:

Consider the coefficient of `(-4)` from `-4x`
`color(brown)(x_("vertex") = (-1/2)(-4) = +2)`

so by substitution:

`color(brown)(y_("vertex") =(2)^2-4(2)+2 = -2)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The `y_("intercept")` is at `x=0`

`color(brown)(=> y_("intercept") = (0)^2-4(0)+2 = 2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Not so quick to find the values for `x_("intercept")` as its factors are non integer values.

Consider standard equation form of: `ax^2+bx+c=0`

and the related `x=(-b+-sqrt(b^2-4ac))/(2a)`

giving: `x=(-(-4) +-sqrt((-4)^2-4(1)(2)))/(2(1))`

`color(green)("I will let you solve that part. Compare your answers to the graph.")`