4. The variable $$ y $$ varies inversely with $$ x $$. When $$ x=5, y=4 $$. Write and graph an inverse variation equation that relates $$ x $$ and $$ y $$.

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1. Since $$ y $$ varies inversely with $$ x $$, the relationship can be written as  
   $$
   y = \frac{k}{x}
   $$
   where $$ k $$ is a constant.

2. Substitute the given values $$ x=5 $$ and $$ y=4 $$ into the equation to find $$ k $$:  
   $$
   4 = \frac{k}{5}
   $$

3. Multiply both sides by 5 to solve for $$ k $$:  
   $$
   k = 4 \times 5 = 20
   $$

4. The inverse variation equation is:  
   $$
   y = \frac{20}{x}
   $$

5. To graph the equation $$ y = \frac{20}{x} $$:

- The graph is a hyperbola with two branches.
   - It has vertical asymptote at $$ x = 0 $$ and horizontal asymptote at $$ y = 0 $$.
   - Plotting several points:
     - When $$ x = 5 $$, $$ y = \frac{20}{5} = 4 $$.
     - When $$ x = 10 $$, $$ y = \frac{20}{10} = 2 $$.
     - When $$ x = -5 $$, $$ y = \frac{20}{-5} = -4 $$.
     - When $$ x = -10 $$, $$ y = \frac{20}{-10} = -2 $$.

The resulting graph shows that as $$ x $$ increases, $$ y $$ approaches 0 from above, and as $$ x $$ decreases, $$ y $$ approaches 0 from below.

Thus, the equation that relates $$ x $$ and $$ y $$ is  
$$
y = \frac{20}{x},
$$  
and its graph is a hyperbola with vertical asymptote $$ x = 0 $$ and horizontal asymptote $$ y = 0 $$.
The inverse variation equation is $$ y = \frac{20}{x} $$. Its graph is a hyperbola with vertical asymptote at $$ x = 0 $$ and horizontal asymptote at $$ y = 0 $$.

The variable varies inversely with . When . Write
and graph an inverse variation equation that relates and .

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