Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y = x^2 - 6x + 4`?

Answer 1

Axis if symmetry: `color(green)(x=3)`
Minimum value: `color(green)(y=-5)`

Explanation:

A quadratic equation opens upward if the coefficient of the squared variable is greater than zero (i.e. it has a minimum value).

The minimum value is attained when the slope of the tangent (as given by the derivative) is equal to zero.

That is the minimum occurs when
`color(white)("XXX")(d(x^2-6x+4))/(dx)=2x-6=0`
or
`color(white)("XXX")x=3`

The axis of symmetry is a vertical line (when `x` is the dependent variable) passing though the minimum.
Therefore the axis of symmetry is `color(green)(x=3)`

The value of the function `y=x^2-6x+4` at the minimum (i.e. when `x=3`) is
`color(white)("XXX")color(green)(y=)(3)^2-6(3)+4=9-18+4=color(green)(-5)`