How do you find x-intercepts, axis of symmetry, maximum or

Step 1
X-intercept is a point where a graph intersects the X-axis, that is a point with coordinates ${\displaystyle (a,0)}$ that satisfies our equation when we substitute ${\displaystyle x=a}$ and ${\displaystyle y=0}$
Therefore, we just have to find a if
${\displaystyle 0=\frac{1}{2}{(a-8)}^2+2}$
This equation, obviously has no real solutions because
${\displaystyle \frac{1}{2}{(a-8)}^2\ge 0}$
because it's a square of some number multiplied by a positive constant, and, therefore,
${\displaystyle \frac{1}{2}{(a-8)}^2+2>0}$
and there are no such values of a when the expression
${\displaystyle \frac{1}{2}{(x-8)}^2+2}$ equals to 0
Step 2
Axis of symmetry of a canonical parabola ${\displaystyle y=x^2}$ is the Y-axis
The parabola $y=\frac{1}{2}{x}^2$ differs from a canonical one only by a multiplier $\frac{1}{2}$, which just squeezes the parabola towards the X-axis by a factor of 2 without changing its axis of symmetry.
Subtracting a positive constant from an argument x in the equation of any function ${\displaystyle y=F\left(x\right)}$ shifts the graph to the right by the value of this constant.
Therefore, the axis of symmetry of
${\displaystyle y=\frac{1}{2}{(x-8)}^2}$
is a straight line parallel to the Y-axis and intersecting the X-axis at point ${\displaystyle x=8}$. The equation of this line is ${\displaystyle x=8}$ (independent of y)
Adding 2 to a function shifts the graph upwards without changing it's axis of symmetry. So, parabola ${\displaystyle \frac{1}{2}{(x-8)}^2+2}$ has a line ${\displaystyle x=8}$ as the axis of symmetry.
Step 3
The minimum point of a canonical parabola ${\displaystyle y=x^2}$ is a point ${\displaystyle (0,0)}$
After squeezing it by a factor of 2 the minimum point does not change its position. After subtracting 8 from the argument the whole graph shifts to the right by 8, and the minimum point shifts by 8 as well, taking a position ${\displaystyle (8,0)}$
After that we add 2 to a function, which shifts the graph upwards by 2, so the minimum point shifts to ${\displaystyle (8,2)}$
Step 4
Y-intercept is a point on the Y-axis where the graph intersects this axis. This is a value of a function when an argument equals to zero.
Substitute ${\displaystyle x=0}$ in the function:
${\displaystyle y=\frac{1}{2}{(0-8)}^2+2}$
from which we derive ${\displaystyle y=34}$
Therefore, a point ${\displaystyle (0,34)}$ is a Y-intercepts.
Answer: On the above graph you see that there is no X-intercepts, the axis of symmetry is a line ${\displaystyle x=8}$, the minimum point is ${\displaystyle (8,2)}$ and the Y-intercepts is 34