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I have recently begun to try and graph most of the functions I see by hand without resorting to the use of my graphing calculator. However, I am having a problem trying to begin to graph [tex]f(x) = \sqrt{x^{2} + 1}[/tex].

The problem is all I really know is that f(0) = 1. Trying to solve for roots shows that there are none. My graphing calculator essentially graphs this as a horizontally stretched out [tex]x^{2} + 1[/tex], and i'm just not sure why.

At first I thought that it would just be a stretched graph similar to that of just [tex]\sqrt{x}[/tex], but now I see that [tex]\sqrt{x^{2} + 1}[/tex] has values on both sides of the y-axis. But what's confusing is that the graph seems to be concave up (like [tex]x^{2}[/tex]) instead of concave up (like [tex]\sqrt{x}[/tex]).

Anyways, does anyone have any recommendations as to how to approach graphing this by hand?

Thanks.