Graphing Without using Calculus $f(x) = \sqrt{x + 2} - \sqrt{x - 2}$

Question

I am trying to solve the following problem:

I can visualize how it looks like "approximately", it's essentially something like $\sqrt{x - 2} - 2$, with the difference that it increases faster. But based on the other parts of this problem it seems a specific shape such as a parabola, hyperbola, circle, or ... should be found to describe the graph of each function. I tried simplifying the function, with no results. Is it even possible to graph this function without taking derivatives?

I also proved that the function can be written like this: $y^2(y^2-4x) = -16$ where $y \ge2$.

Answer 1

Note that the domain of $f$ is $[2,\infty)$. Furthermore, we can write

$$\begin{align} f(x)&=\sqrt{x+2}-\sqrt{x-2}\\\\ &=\frac4{\sqrt{x+2}+\sqrt{x-2}} \end{align}$$

Now, note that $f$ is clearly decreases monotonically from its maximum value $f(2)=2$ and approaches $0$ as $x\to \infty$. In fact, for large $x$, $f(x)$ behaves asymptotically like $\frac{2}{\sqrt x}$.

Answer 2

The graph starts at the point $(2,2)$ and stays above the $x$-axis, decreasing continuously to $0$ as $x$ goes to $\infty.$

You can see all of that from the fact that $$ \begin{align} f(x)=\frac4{\sqrt{x+2}+\sqrt{x-2}} \end{align}$$ Very smooth and nice curve, looks like the tail of $2/{\sqrt x}$

Answer 3

The curves of $\sqrt{x\pm2}$ are identical half-parabolas with an horizontal axis.

The function to be plotted is the vertical difference between them.