Analyze the provided calculations and the graph to understand the function and its properties.

Let's analyze the provided information to understand the function and its graph.

The function seems to be:

$$y = 5 - \frac{x+5}{x^2+5x}$$

First, simplify the function:

$$y = 5 - \frac{x+5}{x(x+5)}$$

We must consider that (x eq 0) and (x eq -5), because these values would make the denominator zero.

If (x eq -5), we can simplify the fraction:

$$y = 5 - \frac{1}{x}$$

This is a hyperbola shifted vertically. Now, let's analyze the given calculations:

• (y(1) = 5 - \frac{1}{1} = 5 - 1 = 4)
• (y(-1) = 5 - \frac{1}{-1} = 5 + 1 = 6)
• (y(\frac{1}{2}) = 5 - \frac{1}{\frac{1}{2}} = 5 - 2 = 3)
• (y(-\frac{1}{2}) = 5 - \frac{1}{-\frac{1}{2}} = 5 + 2 = 7)
• (y(-5) = 5 - \frac{1}{-5} = 5 + \frac{1}{5} = 5\frac{1}{5})
• (y(5) = 5 - \frac{1}{5} = 4\frac{4}{5})
• (y(-4) = 5 - \frac{1}{-4} = 5 + \frac{1}{4} = 5\frac{1}{4})

These calculations confirm the function (y = 5 - \frac{1}{x}) for (x eq -5).

The graph shows a hyperbola with a horizontal asymptote at (y = 5). There is a vertical asymptote at (x = 0).

Now let's plot the points: