I have two functions:
$f(x) = x^2 + 200$
$g(x) = (x + 8)^2$
I am interested in the relationship between the two functions in the region between the two minimums (from x = -8 to x = 0), which looks as follows:
How can I algebraically derive the equation of this relationship? I tried substitution but the results were not as expected (possibly due to real/imaginary numbers?).
If I understand your picture correctly, you want to know, given the value of the blue curve at some $x$-value between -8 and 0, the value of the corresponding red curve. So the input to your function will be a number $s$ between $200$ and perhaps $300$, and the output will be between about $0$ and $60$.
Here goes. Suppose we call the blue value $t$. Then the $x$-coordinate that produced that $t$ value is the one with $$ x^2 + 200 = t \\ x^2 = t - 200 \\ x = \pm\sqrt{t - 200} $$ At this point, you have to notice that in your graphs, $x$ is negative, while the usual definition of square-root is defiend to be the positive one. So you have to write $$ x = -\sqrt{t - 200} $$
Now the point on the red curve corresponding to that $x$ value is $$ (x+8)^2 = (-\sqrt{t-200} + 8)^2 $$ So your function is $$ f(t) = (-\sqrt{t-200} + 8)^2. $$ You can expand that out algebraically, but it doesn't get a lot nicer: $$ f(t) = (t-200) - 16 \sqrt{t-200} + 64 \\ = t - 16 \sqrt{t-200} - 136. $$