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

1 Answer
Jul 25, 2016

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)#