Why graph a function? Please enlighten me as to how graphing a function helps. I can see a graph's utility with simple functions as they instantly give you value of dependent variable. But ignoring them and considering three dimensional complex figures that often arise in higher level math, how does it help you solve the problem if you know that the graph is a knot or a doughnut or in 2D case say the 'U' shape of a parabola. I admire the wonderful shapes but I don't know what to make of them except having a visual treat. How do they help you solve a given problem?
Edit: Take the following graph as example. Some of you may recognize it but for others who don't what information does it convey to you in absence of any equation and how will it help you understand the physical phenomenon. 

Please take me through your mental journey. I am desperate to see the world with same intuitiveness as any one of you do. Thanks again for your time
 A: Here is a simple task that will be tough without graphing: In how many points is the function
$$f(x) = x-\frac{1}{2} \left\lfloor \frac{1}{2} \left(\sqrt{8 x-7}-1\right)\right\rfloor 
   \left(\left\lfloor \frac{1}{2} \left(\sqrt{8 x-7}-1\right)\right\rfloor +1\right)$$
not continuous? From the formula its not obvious but if you look at the graph you will see that the function does something very simple:

A: I suppose it would depend on the problem, or even the type of problem. In some cases, you can't just look at the equation and see what the answer could be, yet after drawing it out, you can clearly see that the line climbs for a while, then levels off at some value. $That$ value may be what you're trying to find.
Other problems you may be looking for symmetry, which you $can$ find algebraically, but it is a lot easier to just draw it and $look$ at it to see if both sides match up.
A: For example if we are talking about $2$-D functions $y=f(x)$ you may use graph to test whether function is one-to-one as it is shown here.
