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.

When graphing rational functions, it's essential to analyze several key characteristics to understand the behavior and shape of the graph. Here are seven primary characteristics to consider: 1. **Domain:** - **Definition:** The set of all possible input values (x-values) for which the function is defined. - **Considerations:** For rational functions, exclude values that make the denominator zero, as these will cause the function to be undefined. 2. **Intercepts:** - **x-intercepts:** Points where the graph crosses the x-axis. Found by setting the numerator equal to zero and solving for x. - **y-intercept:** The point where the graph crosses the y-axis. Found by evaluating the function at x = 0. 3. **Vertical Asymptotes:** - **Definition:** Lines where the function grows without bound (approaches infinity or negative infinity) as x approaches a certain value. - **Identification:** Determine the values of x that make the denominator zero (and are not canceled out by the numerator). 4. **Horizontal and Oblique Asymptotes:** - **Horizontal Asymptote:** Indicates the end behavior of the function as x approaches positive or negative infinity. - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. - If the degrees are equal, the asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator). - **Oblique (Slant) Asymptote:** If the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division to find the oblique asymptote. 5. **Holes (Removable Discontinuities):** - **Definition:** Points where both the numerator and denominator are zero, resulting in an undefined point that can be "filled in." - **Identification:** Factor the numerator and denominator to find and cancel common factors. The x-value that was canceled indicates the hole. 6. **Behavior Near Asymptotes:** - **Vertical Asymptotes:** Analyze how the function behaves as it approaches the asymptote from the left and right (e.g., approaching positive or negative infinity). - **Horizontal/Oblique Asymptotes:** Determine how the function behaves as x approaches infinity or negative infinity, ensuring it aligns with the asymptote. 7. **Intervals of Increase and Decrease (Optional but Useful):** - **Definition:** Regions where the function is rising or falling. - **Analysis:** Use derivatives to determine where the function is increasing or decreasing, which aids in accurately sketching the graph. By systematically analyzing these seven characteristics, you can effectively graph rational functions and understand their key features without the need for plotting numerous points.