Graphing Rational Functions with Transformations (Precalculus - College Algebra 39)

The video begins with an introduction to rational functions and their relation to transformations. The focus is on understanding the interaction between functions and their vertical asymptotes. Two key functions are discussed: f(x) = 1/x, known as the reciprocal function, and f(x) = 1/x^2. The video emphasizes the importance of recognizing the behavior of these functions around their vertical asymptotes, which is crucial for graphing rational functions.

The domain of the function f(x) = 1/x is explored, highlighting that x cannot equal zero due to the function's undefined nature at that point. This creates a vertical asymptote at x = 0. The concept of odd multiplicity for the vertical asymptote is introduced, explaining that it leads to opposite infinities and a non-continuous function across the x-axis. The video also discusses the horizontal asymptote at y = 0 for the function and how end behavior can indicate the presence of a horizontal asymptote.

The video identifies key points for rational functions by evaluating the function at x = 1 and x = -1, resulting in key points (1,1) and (-1,-1). It is noted that f(x) = 1/x is an odd function, which implies symmetry about the origin. The function's graph is illustrated, showing how it interacts with the vertical asymptote and key points, emphasizing the odd nature of the function and its behavior in the first and third quadrants.

The video moves on to discuss the function f(x) = 1/x^2, which has an even vertical asymptote at x = 0 with multiplicity 2. The behavior of even vertical asymptotes is contrasted with odd ones, showing how they exhibit symmetry and approach the same infinity on both sides of the asymptote. The function's horizontal asymptote is also discussed, and key points are identified by evaluating the function at x = 1 and x = -1, maintaining the even nature of the function.

The video script delves into function transformations, specifically focusing on how shifts affect the function's asymptotes. A vertical shift upwards by 3 units is applied to the function f(x) = 1/x^2 + 3, which moves both the vertical and horizontal asymptotes accordingly. The importance of understanding how the function interacts with its asymptotes after a transformation is emphasized, and the video demonstrates how to use key points to graph the transformed function.

The video continues with a discussion on the function g(x) = 1/(x - 1)^2 - 2, highlighting the need to identify shifts for both vertical and horizontal asymptotes. The function undergoes a vertical shift down by 2 units and a horizontal shift to the right by 1 unit. The video explains how these shifts affect the function's graph and how to plot key points in relation to the new asymptotes, maintaining the function's even symmetry across the vertical asymptote.

The video script examines the function h(x) = -2/(x + 3)^2 - 4, noting its basis on the 1/x^2 function with significant transformations applied. The function experiences a downward shift of 4 units and a leftward shift of 3 units. The video demonstrates how to adjust key points according to the function's output multiplier and how to graph the function while respecting its vertical asymptote. The domain of the function is also analyzed to confirm the presence and nature of the vertical asymptote at x = -3.

The video concludes with the function g(x) = -1/(x + 1) + 2, which is based on the reciprocal function 1/x but with several transformations. The function has a vertical shift up by 2 units and a leftward shift of 1 unit. The video explains the change in key points due to the negative coefficient, resulting in a reflection of the original odd function's key points. The graph is plotted with respect to the new asymptotes, and the video uses symmetry to accurately sketch the function's curve. The domain is checked to ensure that the function cannot have x equal to -1, which defines the vertical asymptote at that point.

The final paragraph wraps up the discussion by summarizing the behavior of vertical asymptotes in odd and even functions and how they relate to the transformations of rational functions. The video emphasizes the importance of understanding these concepts for graphing rational functions and sets the stage for future topics such as horizontal and oblique asymptotes.