Let's analyze the given problem step by step. The function provided is:

$f(x) = \frac{-6}{2x - 1}$

We will identify the vertical and horizontal asymptotes, plot points on the graph, and summarize the graph behavior.

1. Vertical Asymptote

Vertical asymptotes occur where the denominator equals zero because the function becomes undefined.

Set the denominator equal to zero: $2x - 1 = 0 \implies x = \frac{1}{2}$

Thus, there is a vertical asymptote at: $x = \frac{1}{2}$

2. Horizontal Asymptote

For rational functions, the horizontal asymptote is determined by comparing the degree of the numerator and denominator:

• The numerator $-6$ has degree 0 (constant).
• The denominator $2x - 1$ has degree 1.

When the degree of the denominator is greater than the numerator, the horizontal asymptote is at: $y = 0$

3. Plotting Two Points on Each Piece of the Graph

To better sketch the graph, let's choose some $x$-values on either side of the vertical asymptote $x = \frac{1}{2}$ and calculate the corresponding $y$-values.

Case 1: $x < \frac{1}{2}$ (Left of Asymptote)

• Choose $x = 0$: $f(0) = \frac{-6}{2(0) - 1} = \frac{-6}{-1} = 6$ Point: $(0, 6)$

• Choose $x = -1$: $f(-1) = \frac{-6}{2(-1) - 1} = \frac{-6}{-2 - 1} = \frac{-6}{-3} = 2$ Point: $(-1, 2)$

Case 2: $x > \frac{1}{2}$ (Right of Asymptote)

• Choose $x = 1$: $f(1) = \frac{-6}{2(1) - 1} = \frac{-6}{2 - 1} = \frac{-6}{1} = -6$ Point: $(1, -6)$

• Choose $x = 2$: $f(2) = \frac{-6}{2(2) - 1} = \frac{-6}{4 - 1} = \frac{-6}{3} = -2$ Point: $(2, -2)$

4. Summary of Key Features

• Vertical Asymptote: $x = \frac{1}{2}$
• Horizontal Asymptote: $y = 0$
• Points to plot:
  • Left of $x = \frac{1}{2}$: $(0, 6)$ and $(-1, 2)$
  • Right of $x = \frac{1}{2}$: $(1, -6)$ and $(2, -2)$

5. Graph Behavior

• As $x \to \frac{1}{2}^-$, $y \to +\infty$.
• As $x \to \frac{1}{2}^+$, $y \to -\infty$.
• As $x \to \pm \infty$, $y \to 0$ (approaching the horizontal asymptote).

Let me know if you'd like further clarification or a step-by-step graphical sketch!

Related Questions:

1. How do you find vertical asymptotes for a rational function?
2. Why does the horizontal asymptote occur at $y = 0$ in this case?
3. What happens to the graph as $x$ approaches infinity or negative infinity?
4. How can you determine whether a rational function has oblique asymptotes?
5. How do vertical asymptotes affect the domain of a function?

Tip:

To quickly determine the horizontal asymptote for a rational function, compare the degrees of the numerator and denominator.