Question

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`?

Answer 1

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]}