Better method for graphing functions with greater complexity

Question

I spent perhaps 30 mins graphing the following function. $$y = \frac{1}{|{2x-x^2}|} + \frac{1}{|{2x+x^2}|}$$

I proceeded by first graphing the first half of the function $y = \frac{1}{|{2x-x^2}|}$.

Then I graphed the second part of the function $y = \frac{1}{|{2x+x^2}|}$.

Then using both graphs I drew a combined version which I used as my answer to the exercise.

This process was fairly time consuming and I am wondering is there a more concise way of doing this? Is there a way of understanding the general shape of such functions like we understand the more basic functions such as $y = \frac{1}{x}$ or $y = x^3$ and then just inputing some key input values to get a more precise description of the graph?

If I could have an intuition of more complex functions that are combinations of the more basic functions it would be much easier to do such graphing exercises. Is there a method for getting such intuition? Is there a more concise way to complete such graphing exercises?

Answer 1

You need to consider four cases:

1. $x<-2$

2. $-2<x<0$;

3. $0<x<2$ and

4. $x>2$.

Answer 2

Abuse that:

$$|x|=x \text { or } -x$$

Figure out when both expressions inside the moduli are positive or negative and graph those cases individually.

Answer 3

Why not just plot it--using software available to all?

And let me strongly recommend anyone who thinks graphing "by hand" is in any way superior to graphing by computer (e.g., because it teaches "fundamentals" or "solidifies understanding"), see this important TED talk.

The OP spent "perhaps 30 min" trying to graph this function by hand and understand it. I spent less than one minute writing the below code. For those who can run Mathematica, try it and leave a comment as to whether or not you understand the underlying function better by interacting with the figure (for 30 minutes!) than by getting out paper and pencil.

Manipulate[
  Plot[1/Abs[a x + x^2] + 1/Abs[b x - x^2], 
  {x, -5, 5}],
 {{a, 2}, 1, 3}, {{b, 2}, 1, 3}]

Answer 4

$2 x - x^2 = x (2 - x) = 0$ when $x=0$ or $2$, and similarly $2 x + x^2 = 0$ when $x = 0$ or $-2$.
$1/|2x - x^2|$ is always positive, goes to $0$ at $\pm \infty$ and to $+\infty$ as $x \to 0$ or $2$, so looks something like this:

(well, exactly like that, since I used software, but you could do a similar rough sketch by hand). $1/|2x + x^2| = 1/|2(-x) - (-x)^2|$ is the reflection of this across the $y$ axis. It goes to $0$ at $\pm \infty$ and to $+\infty$ as $x \to -2$ or $0$. Thus the sum of these looks something like