To graph the function $f(x) = \frac{2x - 6}{-2x + 2}$, let's break down the steps.

Step 1: Determine Vertical Asymptotes

Vertical asymptotes occur when the denominator equals zero, as the function becomes undefined.

$-2x + 2 = 0$ $x = 1$

So, there is a vertical asymptote at $x = 1$.

Step 2: Determine Horizontal Asymptote

For horizontal asymptotes in rational functions, compare the degrees of the numerator and the denominator.

• The degrees of both the numerator (2x) and the denominator (-2x) are the same.
• To find the horizontal asymptote, divide the leading coefficients of the numerator and denominator:

$y = \frac{2}{-2} = -1$

Therefore, there is a horizontal asymptote at $y = -1$.

Step 3: Plot an Additional Point

To better understand the behavior of the function, choose a point not on an asymptote. For example, evaluate $f(0)$:

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

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

Summary

1. Vertical Asymptote: $x = 1$
2. Horizontal Asymptote: $y = -1$
3. Point to Plot: $(0, -3)$

With this information, you can now plot the function:

1. Draw a vertical dashed line at $x = 1$.
2. Draw a horizontal dashed line at $y = -1$.
3. Plot the point $(0, -3)$ on the graph.
4. Sketch the curve, keeping in mind the behavior near the asymptotes.

Would you like further details on how to sketch the curve or additional questions?

Additional Related Questions

1. How do we determine if there is a slant asymptote for rational functions?
2. What is the difference between horizontal and oblique asymptotes?
3. How would this graph change if we added or subtracted a constant from $f(x)$?
4. How do we find intercepts for rational functions like this one?
5. What happens to the graph if the numerator and denominator have the same polynomial terms?

Tip: Always check for intercepts and asymptotes first when sketching rational functions to understand the general shape of the graph.