To graph $f(x) = \frac{x - 6}{x - 2}$ by placing the horizontal and vertical asymptotes, let's analyze the function and determine these asymptotes.

Step 1: Determine Vertical Asymptote

The vertical asymptote occurs where the denominator is zero (since division by zero is undefined). For this function:

$x - 2 = 0 \Rightarrow x = 2$

Thus, there is a vertical asymptote at $x = 2$.

Step 2: Determine Horizontal Asymptote

For rational functions of the form $\frac{ax + b}{cx + d}$, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

In this case, both the numerator and denominator are of degree 1. The leading coefficient in the numerator is 1, and in the denominator, it is also 1. Therefore:

$\text{Horizontal Asymptote: } y = \frac{1}{1} = 1$

So, the horizontal asymptote is $y = 1$.

Step 3: Plot an Additional Point

To get a sense of the graph's shape, let’s pick a point. We can choose $x = 0$ for simplicity:

$f(0) = \frac{0 - 6}{0 - 2} = \frac{-6}{-2} = 3$

So, $(0, 3)$ is a point on the graph.

Summary

1. Vertical Asymptote at $x = 2$.
2. Horizontal Asymptote at $y = 1$.
3. Point $(0, 3)$ as an additional point on the graph.

Now, you can plot these elements on the grid:

• Draw a dashed vertical line at $x = 2$ to represent the vertical asymptote.
• Draw a dashed horizontal line at $y = 1$ to represent the horizontal asymptote.
• Plot the point $(0, 3)$.

Would you like more details on any of these steps, or have any other questions?

Here are 5 questions for further exploration:

1. What happens to $f(x)$ as $x$ approaches the vertical asymptote from both sides?
2. How does the behavior of $f(x)$ change as $x \to \infty$ or $x \to -\infty$?
3. How would the graph of $f(x) = \frac{x - 6}{x - 2}$ change if the numerator or denominator were altered?
4. What would be the effect of adding a constant to the entire function $f(x)$?
5. How can you determine the slant asymptote for a function where the degree of the numerator is one higher than the denominator?

Tip: Always examine both the horizontal and vertical asymptotes to understand the end behavior and restrictions of rational functions before plotting additional points.