https://file.aiquickdraw.com/mathbot/file/1727843680458oy1zr4yj.jpg

Let's break down the steps based on the provided image.

The function given is \( f(x) = \frac{-3}{x} \).

### a) Graph and Make a Table with at least 5 Points:
We will select values of \( x \) and calculate corresponding values of \( f(x) \).

Let's choose the following values for \( x \):
1. \( x = -3 \)
2. \( x = -1 \)
3. \( x = -0.5 \)
4. \( x = 1 \)
5. \( x = 3 \)

**Table of values:**

\[
\begin{array}{|c|c|}
\hline
x & f(x) = \frac{-3}{x} \\
\hline
-3 & 1 \\
-1 & 3 \\
-0.5 & 6 \\
1 & -3 \\
3 & -1 \\
\hline
\end{array}
\]

Now you can plot these points on the graph. Since \( f(x) = \frac{-3}{x} \), this is a hyperbola, and it will approach but never touch the axes (asymptotes).

### b) Constant of Variation:
The constant of variation \( k \) for an inverse variation \( f(x) = \frac{k}{x} \) is the numerator. In this case, \( k = -3 \).

### c) What is \( f(0) \)?
\( f(0) \) is undefined because division by zero is not allowed in mathematics. Therefore, the function does not exist at \( x = 0 \).

### d) Equation of the Vertical Asymptote:
Since \( f(x) \) is undefined at \( x = 0 \), there is a vertical asymptote at \( x = 0 \).

### e) Behavior as \( x \to +\infty \):
As \( x \to +\infty \), \( f(x) = \frac{-3}{x} \) approaches 0 from the negative side. Thus, \( f(x) \to 0^- \).

### f) Behavior as \( x \to -\infty \):
As \( x \to -\infty \), \( f(x) = \frac{-3}{x} \) also approaches 0 from the positive side. Thus, \( f(x) \to 0^+ \).

### g) Equation of the Horizontal Asymptote:
As \( x \to \pm\infty \), \( f(x) \) approaches 0. Therefore, the horizontal asymptote is \( y = 0 \).

### h) Equation of the Axis of Symmetry:
The axis of symmetry for this function is the y-axis because the function is symmetric about the origin. This symmetry can be expressed as:
- \( f(-x) = -f(x) \) (this indicates symmetry with respect to the origin).
 
Let me know if you want me to plot the graph or clarify any part of the solution.

Here are 5 related questions:
1. How do you determine the asymptotes of a rational function?
2. What happens to the value of the function as \( x \) approaches 0 from the left and right?
3. How do inverse variation functions compare to direct variation functions?
4. Why is division by zero undefined?
5. How would the graph change if the numerator were positive instead of negative?

**Tip:** When graphing rational functions, always identify vertical and horizontal asymptotes to understand the behavior at extreme values of \( x \).

Given the function f(x) = -3/x, answer the following questions: a) Graph the function and make a table with at least 5 points. b) What is the constant of variation? c) What is f(0)? d) What is the equation of the vertical asymptote? e) What is the behavior of the function as x approaches positive infinity? f) What is the behavior of the function as x approaches negative infinity? g) What is the equation of the horizontal asymptote? h) What is the equation(s) of the axis of symmetry?

Learn how to graph the rational function f(x) = -3/x with step-by-step explanations on identifying asymptotes, behavior at infinity, and axis of symmetry. Suitable for students in Grades 10-12 studying inverse variation.

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