To graph the function $y = \frac{2 - 3x}{(x - 3)(x - 2)}$, follow these steps:

1. Identify Vertical Asymptotes:

Vertical asymptotes occur where the denominator is equal to zero (since division by zero is undefined).

The denominator is $(x - 3)(x - 2)$. Set each factor equal to zero:

• $x - 3 = 0$ → $x = 3$
• $x - 2 = 0$ → $x = 2$

Therefore, there are vertical asymptotes at $x = 2$ and $x = 3$.

2. Find Horizontal Asymptote:

Horizontal asymptotes describe the behavior of the function as $x \to \infty$ or $x \to -\infty$.

To find the horizontal asymptote, consider the degrees of the numerator and denominator:

• The numerator, $2 - 3x$, is of degree 1 (the highest power of $x$ is 1).
• The denominator, $(x - 3)(x - 2)$, is a quadratic expression, so it is of degree 2.

Since the degree of the denominator is higher than the degree of the numerator, the horizontal asymptote is at $y = 0$.

3. Find the x-intercepts (if any):

The x-intercepts occur where the numerator equals zero (since $y = 0$ when the numerator is zero).

Set the numerator equal to zero: $2 - 3x = 0 \quad \Rightarrow \quad x = \frac{2}{3}$ Therefore, there is an x-intercept at $x = \frac{2}{3}$, or the point $\left(\frac{2}{3}, 0\right)$.

4. Find the y-intercept:

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into the function: $y = \frac{2 - 3(0)}{(0 - 3)(0 - 2)} = \frac{2}{6} = \frac{1}{3}$ Therefore, the y-intercept is at $\left(0, \frac{1}{3}\right)$.

5. Analyze the behavior around the asymptotes:

• As $x \to 3^+$ (from the right of 3), the function approaches $+\infty$, and as $x \to 3^-$ (from the left of 3), the function approaches $-\infty$.
• As $x \to 2^+$, the function approaches $-\infty$, and as $x \to 2^-$, the function approaches $+\infty$.

6. Plot key points and sketch the graph:

• Plot the asymptotes at $x = 2$ and $x = 3$, and the horizontal asymptote at $y = 0$.
• Plot the x-intercept at $x = \frac{2}{3}$ and the y-intercept at $\left(0, \frac{1}{3}\right)$.
• Use the behavior near the asymptotes and at extreme values of $x$ to help sketch the graph.

Summary of Important Points:

• Vertical asymptotes: $x = 2$ and $x = 3$
• Horizontal asymptote: $y = 0$
• X-intercept: $x = \frac{2}{3}$
• Y-intercept: $y = \frac{1}{3}$

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Do you want further details on any of these steps or additional information?

Here are some related questions:

1. How do you determine whether a function has a horizontal asymptote?
2. What happens to the graph of a rational function when there are multiple vertical asymptotes?
3. How can you confirm the end behavior of a rational function as $x \to \infty$?
4. What role do the factors in the denominator play in determining the vertical asymptotes?
5. How would you find the oblique asymptote for a function if the degree of the numerator is one higher than that of the denominator?

Tip: Always analyze the behavior of a function near asymptotes first before plotting other points. This helps you understand the overall shape of the graph.