Homework - Square Root Functions with Exponents

Question

I am doing square root graphs in school and on the homework, one problem was

$$f(x) = \sqrt{x^2-25}$$

I know that there is a horizontal translation of positive 25, but how would I graph the $x^2$ without making an x/y chart?

Answer 1

Notice that the function is defined only for $x \leq -5$ and $5 \leq x$. Also, as $x \to \pm \infty$, the $25$ term becomes insignificant and so $f(x) \to \sqrt{x^2} = |x|$ (try showing this more rigorously).

So the graph starts at $x = \pm 5$ where $f(x) = 0$ and becomes asymptotic to $|x|$ as the magnitude of $x$ increases.