Section $$ 3.6 \# 42 $$ Use the Guidelines for Graphing Rational Functions to list out the seven characteristics of the graph. You do not need to graph the function.

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When graphing rational functions, it's essential to analyze several key characteristics to understand the behavior and shape of the graph. Here are the seven primary characteristics to consider:

1. **Domain**
   - **Definition:** The set of all possible input values (x-values) for which the function is defined.
   - **How to Find:** Identify values of $$ x $$ that make the denominator zero, as these are excluded from the domain.
   - **Example:** For $$ f(x) = \frac{P(x)}{Q(x)} $$, solve $$ Q(x) = 0 $$ to find restrictions.

2. **Intercepts**
   - **X-Intercepts (Zeros):**
     - **Definition:** Points where the graph crosses the x-axis ($$ y = 0 $$).
     - **How to Find:** Set the numerator $$ P(x) = 0 $$ and solve for $$ x $$, ensuring these solutions are within the domain.
   - **Y-Intercept:**
     - **Definition:** The point where the graph crosses the y-axis ($$ x = 0 $$).
     - **How to Find:** Evaluate $$ f(0) $$, provided $$ x = 0 $$ is within the domain.

3. **Vertical Asymptotes**
   - **Definition:** Lines $$ x = a $$ where the function grows without bound (approaches infinity or negative infinity).
   - **How to Find:** Determine the values of $$ x $$ that make the denominator zero but do not cancel with factors in the numerator.
   - **Example:** If $$ f(x) = \frac{(x-2)}{(x-2)(x+3)} $$, there is a vertical asymptote at $$ x = -3 $$ (since $$ x = 2 $$ is a hole).

4. **Horizontal or Oblique Asymptotes**
   - **Horizontal Asymptotes:**
     - **Definition:** Lines $$ y = b $$ that the graph approaches as $$ x $$ approaches $$ \pm\infty $$.
     - **How to Find:** Compare the degrees of the numerator and denominator:
       - If degree of numerator < degree of denominator: $$ y = 0 $$.
       - If degree of numerator = degree of denominator: $$ y = $$ ratio of leading coefficients.
   - **Oblique (Slant) Asymptotes:**
     - **Definition:** Linear asymptotes of the form $$ y = mx + b $$ when the degree of the numerator is one more than the denominator.
     - **How to Find:** Perform polynomial long division to find $$ y = mx + b $$.

5. **Holes (Removable Discontinuities)**
   - **Definition:** Points where the function is not defined due to a common factor in the numerator and denominator that cancels out.
   - **How to Find:** Identify and cancel common factors, then mark the canceled $$ x $$-value as a hole.
   - **Example:** In $$ f(x) = \frac{(x-1)}{(x-1)(x+2)} $$, there's a hole at $$ x = 1 $$.

6. **Behavior Near Asymptotes**
   - **Vertical Asymptotes:** Determine whether the graph approaches $$ +\infty $$ or $$ -\infty $$ as $$ x $$ approaches the asymptote from the left and right.
   - **Horizontal/Oblique Asymptotes:** Confirm the end behavior aligns with the asymptote as $$ x $$ approaches $$ \pm\infty $$.

7. **Additional Features**
   - **Symmetry:**
     - **Even Function:** Symmetric about the y-axis.
     - **Odd Function:** Symmetric about the origin.
     - **No Symmetry:** Neither even nor odd.
   - **Intervals of Increase/Decrease:** Determine where the function is rising or falling.
   - **Local Extrema:** Identify any local maximum or minimum points.

**Summary Checklist:**
1. Determine the **domain**.
2. Find all **intercepts** (x and y).
3. Identify **vertical asymptotes**.
4. Determine **horizontal or oblique asymptotes**.
5. Locate any **holes** in the graph.
6. Analyze the **behavior near asymptotes**.
7. Examine **additional features** like symmetry and intervals of increase/decrease.

By systematically analyzing these characteristics, you can accurately sketch the graph of a rational function without actually plotting the points.
To graph a rational function, identify the following seven characteristics: 1. **Domain:** All x-values where the function is defined. 2. **Intercepts:** Points where the graph crosses the x and y-axes. 3. **Vertical Asymptotes:** Lines where the function grows without bound. 4. **Horizontal or Oblique Asymptotes:** Lines the graph approaches as x increases. 5. **Holes:** Points where the function is undefined due to common factors. 6. **Behavior Near Asymptotes:** How the graph approaches asymptotes from both sides. 7. **Additional Features:** Symmetry, increasing/decreasing intervals, and local extrema. By analyzing these, you can understand the graph's shape and key features without plotting points.

Section \( 3.6 \# 42 \) Use the Guidelines for Graphing Rational Functions to list out the seven characteristics of the graph. You do not need to graph the function.

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