How to improve visualization skills (Graphing)

Question

Okay, so my problem is, that I have difficulty visualizing graphs of functions.
For example, if we have to calculate the area bounded by multiple curves, I face difficulty in visualizing that how the graph would look like. And in Integral Calculus, it is very important to plot the graph in some problems, so how can I improve my visualization skills? What can I implement in my daily math problem solving to improve these skills?
Edit:I am not talking about normal functions, simple ones like Signum function, absolute value function etc. I am talking about more complex ones like \begin{equation*} y=e^{|x-1|+|x+2|}. \end{equation*}

Answer 1

To better grasp the behavior of complex functions, in the first place you must master the behavior of the classical ones, like the powers ($x^d$, for all $d\in\mathbb R$), the polynomials, the exponential, the trigonometric functions and their inverses. You must know by heart their range and their domain, their behavior at $\pm\infty$, their asymptotes and their remarkable points like extrema and inflections, their periodicity.

You must also have some understanding of what occurs when you combine functions, by addition, subtraction, multiplication or division. What about the asymptotic behavior, what about remarkable points (are they preserved, do they move...)

You should also be able to relate the slope (first derivative) to the graph itself, and in particular identify monotonous sections. Also by the second derivative, increasing/decreasing growth.

You should also be able to select "interesting points" and compute function values there. All of this takes a lot of practice and experience.

Let us look at your given example, $e^{|x-1|+|x+2|}$.

We first notice that this is an exponential function applied to the argument $|x-1|+|x+2|$. The exponential function has a pretty simple behavior: it is monotonic, growing from $0$ at $-\infty$, to $\infty$ at $\infty$. The growth is extremely fast. There are no extrema, no inflections, an horizontal asymptote. A simple point is $(0,1)$.

Let us now focus on the argument. It is the sum of two absolute values. The absolute value function is piecewise linear, with slope $-1$, then $1$ at the point $(0,0)$. Its graph is two half-lines meeting at the origin. In the case at hand, we have a sum of two such functions, horizontally translated by $1$ and $-2$ units respectively. When you add these, you will obtain a piecewise linear function, with slopes $-2$, $0$ and $2$. The section with slope $0$ is the constant $3$, and this is the minimum value.

Now if you "pass" this function to the exponential, the graph will get deformed non-linearly with large values becoming larger and larger. You will get a section of the decreasing exponential $e^{-2x}$ from $-\infty$ to $-2$, then the constant $e^3$, then the increasing exponential $e^{2x}$, from $1$ to $\infty$.

On the plot below, the $y$ axis has been compressed to make the exponential behavior visible. (In blue, the argument of the exponential.)

Answer 2

A friend of mine sent me a link to graph paper site: http://www.printablepaper.net/category/graph

gives you a choice of squares per inch and whether there are bolder lines each inch. Draw stuff. Do not imagine drawing things. Actually do so. If you want to solve $3 |x| = e^x,$ print out some graph paper, draw $y = 3|x|$ and $y=e^x$ on the same page and see what happens.

There is a tradition in, let us call it, neuroscience, possibly a small minority opinion, that we have intelligence precisely because we have hands; one aspect of this is http://en.wikipedia.org/wiki/Homo_faber I see lots of students on this site who have no ability to visualize in either two or three dimensions because they have never drawn any pictures or built any models of polyhedra. Part of this is that software such as mathcad took over early in many engineering and architectural fields; the people involved are the poorer for it. This has some references: http://www.waldorfresearchinstitute.org/pdf/Hand-Movements-Create-Intelligence.pdf