A function is symmetric with respect to the y-axis if f(-x)=f(x), because this way, all y-axis-symmetric points "belong" to the function. For example, if the point (1,4) is a point of the function (which is the same as saying that f(1)=4), then f(-1)=f(1)=4, and, hence, the point (-1,4) belongs to the function too. And (-1,4) and (1,4) are symmetric with respect to the y-axis.
To know if a function is symmetric with respect to the y-axis, you only have to compare f(x) and f(-x). If they are equal for all x, then the function is symmetric. Otherwise, the function is not symmetric.
In this exercise, you have:
1) f(-x)=|-x|=|x|=f(x) => f is symmetric
2) f(-x)=|-x|+3=|x|+3=f(x) => f is symmetric
3) f(-x)=|-x+3|, but this is not always equal to |x+3| => f is NOT symmetric
4) f(-x)=|-x|+6=|x|+6=f(x) => f is symmetric
5) f(-x)=|-x-6|, but this is not always equal to |x-6| => f is NOT symmetric
6) f(-x)=|-x+3|-6, but this is not always equal to |x+3|-6 => f is NOT symmetric
You can also reason this way: f(x)=|x| is symmetric with respect to the y-axis. It's easy to see that f(-x)=f(x). Now, interpret the other functions as translations in the y-direction or in the x-direction, or both.
f(x-m) is a function like f but translated to the right m units.
f(x)+n is a function like f but translated up n units.
In the next paragraph, symmetric means always symmetric with respect to the y-axis.
If you translate a symmetric function vertically, the result is still symmetric. But if you translate it horizontally, the result is no longer y-axis symmetric (unless the original function was constant or periodic).
Functions 2) and 4) are vertical translations of a symmetric function: they are symmetric. Functions 3) and 5) are horizontal translations of a symmetric function: they are not symmetric. Function 6) is translated both vertically and horizontally: it is not symmetric.
