The task is to find the common point of the graphs $K$ and $G$ represented by the functions $f(x)$ and $g(x)$, respectively:

Given functions:

1. $f(x) = -x^{\frac{1}{2}}$, with $x \geq 0$
2. $g(x) = 3x^{\frac{1}{2}} - 20$, with $x \geq 0$

To find the common point, we need to set $f(x) = g(x)$ and solve for $x$.

Solution:

1. Set $f(x) = g(x)$: $-x^{\frac{1}{2}} = 3x^{\frac{1}{2}} - 20$

2. Add $x^{\frac{1}{2}}$ to both sides: $0 = 4x^{\frac{1}{2}} - 20$

3. Rearrange to isolate $x^{\frac{1}{2}}$: $4x^{\frac{1}{2}} = 20$

4. Divide by 4: $x^{\frac{1}{2}} = 5$

5. Square both sides to solve for $x$: $x = 25$

6. Substitute $x = 25$ back into either $f(x)$ or $g(x)$ to find the $y$-coordinate.

   Using $f(x) = -x^{\frac{1}{2}}$: $f(25) = -\sqrt{25} = -5$

Thus, the common point of the graphs $K$ and $G$ is: $(25, -5)$

Would you like further details or explanations on this solution?

Here are related questions you might find helpful:

1. How do you find the intersection point of two linear functions?
2. What are the common methods to solve equations with fractional exponents?
3. How can you check if two functions intersect only at a single point?
4. How does changing the value of the constant in a function like $g(x)$ affect the intersection?
5. What are some applications of finding intersections of graphs in real-world problems?

Tip: Always double-check by substituting back to verify that your solution satisfies both functions.