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 the seven primary characteristics to consider: 1. **Domain:** - **Determine the Domain:** Identify all real numbers for which the function is defined. For rational functions, exclude any values of \( x \) that make the denominator zero since division by zero is undefined. 2. **Intercepts:** - **x-Intercepts (Zeros):** Find the values of \( x \) that make the numerator zero (provided these values do not also make the denominator zero). These points are where the graph crosses the x-axis. - **y-Intercept:** Evaluate the function at \( x = 0 \) to find the point where the graph crosses the y-axis. 3. **Vertical Asymptotes:** - **Identify Vertical Asymptotes:** Determine the values of \( x \) that make the denominator zero but do not cancel out with factors in the numerator. These lines represent vertical asymptotes where the function approaches infinity or negative infinity. 4. **Holes (Removable Discontinuities):** - **Find Holes:** If a factor cancels out in both the numerator and the denominator, it creates a hole in the graph at that specific \( x \)-value. This is a point of removable discontinuity. 5. **Horizontal or Oblique Asymptotes:** - **Determine End Behavior:** - **Horizontal Asymptote:** Compare the degrees of the numerator and denominator. - If the degree of the numerator is less than the denominator, the horizontal asymptote is \( y = 0 \). - If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. - **Oblique (Slant) Asymptote:** If the degree of the numerator is exactly one more than the denominator, there is an oblique asymptote, which can be found using polynomial long division. 6. **Behavior Near Asymptotes:** - **Analyze Function Behavior:** Examine how the function behaves as it approaches the vertical and horizontal (or oblique) asymptotes. Determine whether it approaches infinity or negative infinity near vertical asymptotes and how it settles near horizontal or oblique asymptotes. 7. **Additional Points and Graph Shape:** - **Plot Key Points:** Choose additional \( x \)-values within different intervals defined by the asymptotes to plot points and understand the curvature of the graph. - **Determine Symmetry and Intervals:** Assess whether the function is symmetric about the y-axis, x-axis, or origin, and analyze the intervals between asymptotes to sketch the overall shape accurately. By systematically evaluating these seven characteristics, you can comprehensively understand and graph rational functions without necessarily plotting every point.