Graphing Rational Functions With Vertical, Horizontal & Slant Asymptotes, Holes, Domain & Range

This paragraph introduces the topic of graphing rational functions, focusing on identifying asymptotes—both horizontal and vertical—as well as slant and oblique ones. It also discusses how to determine the domain and range of a function. The parent function, y = 1/x, serves as an example to illustrate these concepts. The domain is explained as all possible x-values excluding vertical asymptotes and holes, while the range is all possible y-values excluding horizontal asymptotes and holes. The function's graph shows behavior in the upper right and lower left corners, with specific asymptotes and domain and range definitions provided.

The paragraph delves into the details of graphing the function y = 1/x^2, starting with finding its vertical asymptote at x = 0 by setting the denominator to zero. It explains that the horizontal asymptote remains y = 0 due to the 'bottom-heavy' nature of the function. The graph's behavior is described, emphasizing the positive values since squaring a number cannot yield a negative result. The domain and range are discussed, noting that the domain remains the same as the parent function, but the range starts from y = 0 to positive infinity, reflecting the function's behavior above the x-axis.

This section discusses how to graph the function f(x) = 1/(x-1) by identifying its vertical asymptote at x = 1, achieved by setting the denominator to zero. The horizontal asymptote is still y = 0, as the function remains bottom-heavy. The paragraph provides a step-by-step guide to sketching the graph, including choosing strategic points to illustrate the function's behavior near the vertical asymptote and on both sides of it. It also includes a brief on how to find points for an accurate graph and how the left side of the graph mirrors the right side, with specific x-values chosen to demonstrate this.

The paragraph explains the process of graphing the function y = 1/(x+2) - 3, starting with finding the vertical asymptote at x = -2 and the horizontal asymptote at y = -3. It describes how to plot these asymptotes and suggests plugging in points to sketch the graph accurately. The function's graph is characterized by its behavior around the asymptotes, and specific points are calculated to illustrate this. The domain and range are then discussed, with the domain excluding the vertical asymptote and the range reflecting the function's limits between negative infinity and the horizontal asymptote.

This section examines the impact of adding a negative sign to rational functions, demonstrating how it reflects the graph over the x-axis. It uses the function y = -1/(x+2) + 3 as an example to show how the vertical asymptote shifts left by two units and the horizontal asymptote is raised by three units. The paragraph also explains that the negative sign causes the graph to be below the new horizontal asymptote of y = 3 and provides a rough sketch of the graph based on points calculated on either side of the vertical asymptote.

The paragraph explores how to determine horizontal asymptotes for various rational functions, depending on whether the function is bottom-heavy, has the same degree for numerator and denominator, or has a numerator degree exceeding the denominator's. It explains the rules for calculating horizontal asymptotes in each case, including the effects of constants outside the function. The section also provides examples of finding horizontal asymptotes for given functions and touches on the concept of slant asymptotes for functions where the numerator's degree exceeds the denominator's by one.

This section introduces the concept of slant or oblique asymptotes for rational functions where the degree of the numerator exceeds that of the denominator by one. It demonstrates the process of finding a slant asymptote through long division using the example y = 2(x-2)/(x-3). The paragraph explains how to perform the division to obtain the equation of the slant asymptote and discusses the vertical and horizontal asymptotes of the function, including how to find x-intercepts and the behavior of the graph near these asymptotes.

The paragraph discusses the process of graphing rational functions with slant asymptotes, starting with factoring the numerator and denominator to identify vertical asymptotes and holes. It uses the function f(x) = (3x^2 + 9x - 12)/(x^2 + x - 2 - 4) as an example, showing how to simplify the expression by factoring and finding the GCF. The horizontal asymptote is calculated from the leading coefficients, and the vertical asymptote is identified. The paragraph also describes how to find x-intercepts and y-intercepts and provides a step-by-step guide to graphing the function, including the shape of the graph around the asymptotes and holes.

This section focuses on graphing functions with slant asymptotes by using long division to find the equation of the asymptote. The example function 2x^2 + 6x - 8/x - 2 is used to demonstrate this process. The paragraph explains how to perform long division to obtain the slant asymptote y = 2x + 10 and how to factor the numerator to find vertical asymptotes and x-intercepts. It also discusses the importance of knowing the highest and lowest points of the graph, which can be found using a graphing calculator, and how to plot these points to sketch the function's graph.

The final paragraph wraps up the discussion on graphing rational functions, summarizing the key points covered in the video. It emphasizes the importance of understanding how to identify horizontal, vertical, and slant asymptotes, as well as locating holes in the graph. The paragraph also highlights the process of determining the domain and range of these functions, mentioning the use of graphing calculators for complex graphs. The video concludes with a reminder of the steps involved in graphing rational functions and a thank you note to the viewers.