Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y=x^2-8x+3`?

Answer 1

This form has a vertical axis of symmetry through the vertex, opening upward. Find the vertex to get the minimum (the `y` coordinate) and the axis of symmetry (`x =` the `x` coordinate)

Explanation:

Any parabolic equation in the form:
`color(white)("XXX")y=px^2+qx+r`
has a vertical axis of symmetry and
`color(white)("XXX")`if `p>0` opens upward;
`color(white)("XXX")`if `p<0` opens downward.

Since
`color(white)("XXX")y=x^2-8x+3`
has an implied value of `p=+1`
it opens upward `rArr` its vertex is its minimum.

Converting `y=x^2-8x+3` into vertex form:

Complete the square:
`color(white)("XXX")y=x^2-8xcolor(green)(+16) +3 color(green)(-16)`

`color(white)("XXX")y=(x-4)^2-13`
which is the vertex form with the vertex at `(4,-13)`

The minimum value is `(-13)`
and the axis of symmetry is `x=4`.

For verification purposes here is the graph of `y=x^2-8x+3`
graph{x^2-8x+3 [-15.56, 16.48, -15.01, 1.01]}