How do you find the axis of symmetry, graph and find the maximum or minimum value of the function #y-2=2(x-3)^2#?

1 Answer
Mar 19, 2017

see explanation.

Explanation:

#" Express " y-2=2(x-3)^2" in the form"#

#rArry=2(x-3)^2+2#

The equation of a parabola in #color(blue)"vertex form"# is.

#color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))#
where (h ,k) are the coordinates of the vertex and a is a constant.

#y=2(x-3)^2+2" is in this form"#

#"here " h=3" and " k=2#

#rArr" vertex "=(3,2)#

To determine min/max consider the value of a

#• a>0rArr" minimum " uuu#

#• a<0rArr" maximum " nnn#

#"here " a=2rArr" minimum"#

The axis of symmetry passes through the vertex and is vertical with equation #color(blue)(x=3)#

The minimum value at the vertex is y = 2

#color(blue)"Intercepts"#

#x=0toy=2(-3)^2+2=20larrcolor(red)" y-intercept"#

#y=0to2(x-3)^2=-2#

#rArr(x-3)^2=-1# which has no real solutions and therefore graph does not cross the x-axis.
graph{(y-2x^2+12x-20)(y-1000x+3000)=0 [-40, 40, -20, 20]}