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