How to graph a rational function using 6 steps

The script begins with an explanation of vertical asymptotes, instructing the audience to set the denominator of a function equal to zero to find them. The example given involves factoring and solving for x, resulting in asymptotes at x = 3 and x = -2, which are represented with dashed lines on the graph. The concept of domain is also discussed, clarifying that the values making the denominator zero are not included in the domain, hence not crossing the asymptotes. The script then moves on to horizontal asymptotes, explaining their determination by comparing the degrees of the numerator and the denominator. In this case, since the degree of the numerator is less than the degree of the denominator, a horizontal asymptote at y = 0 is identified. The process of finding x and y-intercepts is also detailed, with the x-intercept found by setting the function equal to zero and solving for x, yielding x = 1. The y-intercept is found by substituting x = 0 into the function, resulting in a y-intercept at 1/6. The absence of slant asymptotes in this function is noted, and the importance of understanding asymptotes as values the graph approaches is emphasized.

This paragraph continues the mathematical discussion by addressing the graphing of functions, focusing on the concept of symmetry. The function f(-x) is evaluated to check for y-axis symmetry, but it is determined that the function does not exhibit symmetry since f(-x) does not simplify to the same or opposite form as the original function. The script then emphasizes the importance of plotting points to understand the graph's shape, suggesting the use of a calculator or table function for efficiency. Specific x-values are evaluated to demonstrate how to find corresponding f(x) values, which are then used to plot points on the graph. The instructor also corrects a mistake made during the explanation, reinforcing the importance of accuracy in mathematical processes. The paragraph concludes with a brief mention of even functions and the advantage they offer in graphing by allowing for mirroring the graph across the y-axis.

The final paragraph of the script involves the instructor plotting additional points on the graph to demonstrate its shape and confirming the values of the function at specific x-values. There is an interactive element where a student corrects the instructor, highlighting the collaborative nature of the learning process. The instructor acknowledges the correction and proceeds to plot the points, which helps visualize the graph's behavior. The paragraph concludes with a brief interaction with a student named Zach, ensuring understanding and engagement in the lesson.