Answer
See graph and explanations.
Work Step by Step
Step 1. Given $y=\frac{2x}{x+5}=\frac{2(x+5)-10}{x+5}=2-\frac{10}{x+5}$
Step 2. It is easier to graph the function by translating the known function $f(x)=-\frac{1}{x}$, where $f'(x)=\frac{1}{x^2}\lt0$ and thus increasing over all its open intervals. The asymptotes are $x=0$ and $y=0$ for this function.
Step 3. To graph the given function from $f(x)=-\frac{1}{x}$, first stretch vertically by a factor of $10$ to get $-\frac{10}{x}$, then shift $5$ units to the left to get $-\frac{10}{x+5}$, and finally shift up $2$ units to get $y=2-\frac{10}{x+5}$ as shown in the figure.
Step 4. The asymptotes will also shift to become $x=-5$ and $y=2$.
