To solve and graph the given rational function $f(x) = \frac{2}{x-1}$, we need to determine its key features: vertical asymptote, horizontal asymptote, and points on the graph.

Step 1: Vertical Asymptote

The vertical asymptote occurs where the denominator is zero because the function is undefined there.

Set $x - 1 = 0$:
$x = 1$
Thus, there is a vertical asymptote at $x = 1$.

Step 2: Horizontal Asymptote

For rational functions of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials:

• If the degrees of $P(x)$ and $Q(x)$ are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

Here:

• The numerator $2$ is degree 0 (constant), and the denominator $x-1$ is degree 1.

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

Step 3: Plot Two Points

To plot points, choose values of $x$ on either side of the vertical asymptote $x = 1$:

1. When $x = 0$: $f(0) = \frac{2}{0 - 1} = -2$ Point: $(0, -2)$

2. When $x = 2$: $f(2) = \frac{2}{2 - 1} = 2$ Point: $(2, 2)$

These points give you reference values on both sides of the asymptote.

Step 4: Draw the Graph

• Draw the vertical asymptote $x = 1$ as a dashed vertical line.
• Draw the horizontal asymptote $y = 0$ as a dashed horizontal line.
• Plot the points $(0, -2)$ and $(2, 2)$.
• Sketch the curve approaching the asymptotes as $x \to 1^-$ and $x \to 1^+$, and as $x \to \pm\infty$. The curve will look like two distinct branches.

Final Observations:

• The curve decreases toward negative infinity as $x \to 1^-$.
• The curve increases toward positive infinity as $x \to 1^+$.
• The graph levels off at $y = 0$ as $x \to \pm\infty$.

Would you like me to explain how to sketch this step-by-step on the graph or provide further details?

Related Questions:

1. How do you identify the vertical and horizontal asymptotes of any rational function?
2. What happens when the degree of the numerator is larger than the denominator?
3. How can you determine slant asymptotes for a rational function?
4. Why are vertical asymptotes undefined at certain $x$-values?
5. How do you calculate intercepts for rational functions like this one?

Tip: Always test a few points on either side of the vertical asymptote to sketch a smooth curve!