For $x$ and $k$ real numbers, determine the values of $k$ for which the graphs of $f(x)=x^2-4$ and $g(x)=2|x|+k$ do not intersect For $x$ and $k$ real numbers, determine the values of $k$ for which the graphs of $f(x)=x^2-4$ and $g(x)=2|x|+k$ do not intersect.
For $x$ and $k$ real numbers, determine the values of $k$ for which the graphs of $f(x)=x^2-4$ and $g(x)=2|x|+k$ have exactly 2 intersections.
I graphed them on Desmos and found that when k is smaller than -5 they don’t have any intersections and when k is greater or equal to -5 they’ll have two intersections, but I really have no idea what to begin with when it comes to handwriting.

If the 2 functions don't have an intersection, then we must have that $f(x) \ne g(x)$ for all $x$, i.e. $x^2-4 \ne 2|x|+k$ which gives that $x^2 -2|x| -4-k \ne 0$.
Do you know what to do with the absolute value? Try to take 2 cases, if $|x|= x$ and if $|x|=-x$
And find the discriminant of this quadratic equation, when is it negative so that it doesn't have roots?
