Question

Part 1.] Does the graph have even symmetry, odd symmetry, or neither?

Answer 1

Part 1
The graph has even symmetry.  You can see that because it is symmetric with respect to the y-axis.
Functions that have even symmetry have the following property:
$f(x)=f(-x)$
Part 2
To answer this we can simply check if the property we mentioned earlier holds for this function.
$sin(\frac{\pi}{2})\ne sin(-\frac{\pi}{2})$
We can see that sine does not have even symmetry. 
In fact, sine function has the following property: 
$sin(x)=-sin(x)$
This is called odd symetry.
Part 3
Take a look at the function that you attached in the picture. We know that function has even symmetry. 
Reflection over x-axis and 180° rotation around the origin would give us -f(x). We would not end up with the same function, so these two are out. 
90° rotation around the origin would mean we swapped x and y so that one is out too. Reflection over the line y=x is a property of functions that have an odd symmetry.
We are left with reflection around y-axis and, as mentioned before, this is the property of evenly symmetric functions.