Cancel Done $$ \overleftrightarrow{\Delta B} $$ RESET (a) 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 of the graph thoroughly. Here are the seven primary characteristics to consider:

1. **Domain:**
   - **Definition:** The set of all real numbers $$ x $$ for which the function $$ f(x) = \frac{P(x)}{Q(x)} $$ is defined.
   - **How to Find:** Identify values of $$ x $$ that make the denominator $$ Q(x) = 0 $$ since division by zero is undefined. Exclude these values from the domain.

2. **Intercepts:**
   - **x-Intercepts (Zeros):**
     - **Definition:** Points where the graph crosses the x-axis ($$ f(x) = 0 $$).
     - **How to Find:** Solve $$ P(x) = 0 $$ while 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 in the domain.

3. **Vertical Asymptotes:**
   - **Definition:** Lines $$ x = a $$ where the function grows without bound as $$ x $$ approaches $$ a $$.
   - **How to Find:** Determine values of $$ x $$ that make the denominator zero and are not canceled out by the numerator (i.e., not removable discontinuities).

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 the denominator.
       - If the degree of $$ P(x) $$ < degree of $$ Q(x) $$, the horizontal asymptote is $$ y = 0 $$.
       - If the degrees are equal, the asymptote is $$ y = \frac{	ext{leading coefficient of } P}{	ext{leading coefficient of } Q} $$.
   - **Oblique (Slant) Asymptotes:**
     - **Definition:** Linear asymptotes that occur when the degree of $$ P(x) $$ is one more than the degree of $$ Q(x) $$.
     - **How to Find:** Perform polynomial long division to find the asymptote.

5. **Holes (Removable Discontinuities):**
   - **Definition:** Points where both the numerator and denominator are zero, resulting in a canceled factor.
   - **How to Find:** Identify common factors in $$ P(x) $$ and $$ Q(x) $$ and determine the corresponding $$ x $$-values. These points are excluded from the graph and are typically marked as holes.

6. **Behavior Near Asymptotes:**
   - **Vertical Asymptotes:**
     - **Behavior:** Determine whether the function approaches $$ +\infty $$ or $$ -\infty $$ as $$ x $$ approaches the asymptote from the left and right.
   - **Horizontal/Oblique Asymptotes:**
     - **Behavior:** Analyze how $$ f(x) $$ approaches the asymptote as $$ x $$ moves towards $$ \pm\infty $$.

7. **End Behavior:**
   - **Definition:** The behavior of the graph as $$ x $$ approaches $$ \infty $$ or $$ -\infty $$.
   - **How to Find:** Relate this to the horizontal or oblique asymptotes determined earlier. It describes how the function behaves at the extreme ends of the graph.

By systematically analyzing these seven characteristics, you can gain a comprehensive understanding of the rational function's graph without the need to plot each point manually.
When graphing rational functions, consider the following seven key characteristics: 1. **Domain:** All real numbers $$ x $$ where the function is defined (excluding values that make the denominator zero). 2. **Intercepts:** - **x-Intercepts:** Points where the graph crosses the x-axis ($$ f(x) = 0 $$). - **y-Intercept:** The point where the graph crosses the y-axis ($$ x = 0 $$). 3. **Vertical Asymptotes:** Lines $$ x = a $$ where the function grows without bound as $$ x $$ approaches $$ a $$. 4. **Horizontal or Oblique Asymptotes:** Lines $$ y = b $$ that the graph approaches as $$ x $$ approaches $$ \pm\infty $$. 5. **Holes (Removable Discontinuities):** Points where both the numerator and denominator are zero, resulting in canceled factors. 6. **Behavior Near Asymptotes:** How the function approaches asymptotes from the left and right. 7. **End Behavior:** The function's behavior as $$ x $$ approaches $$ \infty $$ or $$ -\infty $$. By analyzing these characteristics, you can understand the graph's overall shape and key features without plotting individual points.

Cancel Done \( \overleftrightarrow{\Delta B} \) RESET (a) 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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