Question 786276
1) The function {{{y = 5(x-3)^2+8}}} or {{{f(x) = 5(x-3)^2+8}}}
takes the value {{{y=8}}} when {{{x=3}}} <---> {{{x-3=0}}}
For all other values of {{{x}}},
{{{x-3<>0}}}, {{{(x-3)^2>0)}}}, {{{5(x-3)^2>0)}}}, and {{{y = 5(x-3)^2+8>8}}}
So the point (3,8) (with {{{x=3}}} and {{{y=8}}} is a minimum, because the function's value is greater everywhere else, and it takes the value y=8 only for x=3.
 
2) In the function {{{x}}} appears only in {{{(x-3)^2=(abs(x-3))^2}}}, and
{{{abs(x-3)}}} is the horizontal distance to the vertical line {{{x=3}}}.
The value of {{{abs(x-3)}}} determines the value of {{{y}}}, so that any two points on opposite sides, but at the same horizontal distance from {{{x=3}}} have the same {{{y}}}. If the line {{{x=3}}} were a mirror, the points of the graph on one side would be the mirror image of the points on the other side.
{{{x=3}}} is the equation of the "line of symmetry" or "axis of symmetry."
 
3) I do not know what you meant by "The value of x at which the maximum or minimum of the axis of symmetry."
 
4) The x co-ordinate of the minimum or maximum point is {{{x=3}}} as explained in 1).
 
5. The range of the function is [8,infinity) {{{y}}} can get as large as you want. There are no bounds for {{{y}}}, except tha it must be {{{y>=8}}}. It cannot be less than 8, but it can be more than any number you can name.

6. The maximum or minimum value. As said for 1) the minimum value for y is 8.
 
7. The y co-ordinate of the maximum or minimum point. y=8, as said before.

{{{graph(300,300,-4,10,-5,45,(x-3)^2+8)}}}