Graphing Rational Functions Using Transformations With Vertical and Horizontal Asymptotes

This paragraph introduces the concept of graphing rational functions using transformations. It starts with the basic function 1/x, explaining that its graph exists in quadrants one and three, and identifying the vertical asymptote at x=0 and the horizontal asymptote at y=0. The paragraph then compares the graph of 1/x with -1/x, noting the reflection over the origin and the symmetry about the origin, which indicates that they are odd functions. The domain and range for 1/x are discussed, excluding x=0 from the domain and y=0 from the range. The paragraph also explores the graphs of 1/x^2 and -1/x^2, which are even functions with vertical and horizontal asymptotes at x=0 and y=0, respectively. The domain for these functions is all real numbers except zero, and their ranges are from 0 to infinity (not including 0) and from negative infinity to 0 (also not including 0), respectively.

The second paragraph delves into how to graph rational functions that have been transformed through shifts and reflections. It discusses the function 1/(x+2), which is a horizontal shift of the basic 1/x function two units to the left, with a vertical asymptote at x=-2. The domain and range are analyzed, with the domain being all real numbers except x=-2 and the range being all real numbers except y=0. The paragraph also covers the function -1/(x-3), which includes a horizontal shift three units to the right and a reflection over the x-axis, resulting in a domain from negative infinity to 3 and from 3 to infinity, and a range from negative infinity to infinity, excluding y=0. The transformations are summarized with the function 1/x+2, which has a vertical shift up two units and retains the vertical asymptote at x=0, leading to a domain of all real numbers and a range from negative infinity to 0 and from 0 to infinity, excluding y=0.

This paragraph focuses on plotting asymptotes and graphing rational functions such as 1/x+2 and 1/x^2-3. For 1/x+2, the horizontal asymptote is y=2, and the vertical asymptote is the y-axis (x=0). The domain is from negative infinity to 0 and from 0 to infinity, while the range is from negative infinity to 2 and from 2 to infinity, excluding y=2. The function 1/x^2-3 has a horizontal asymptote at y=-3 and a vertical asymptote at x=0. The graph is symmetric about the y-axis and is shifted down 3 units from 1/x^2. The domain excludes x=0, and the range is from negative 3 to infinity. The paragraph also challenges the reader to sketch the graph of 3-1/(x+2), identifying its asymptotes and domain and range using interval notation.

The final paragraph provides a deeper analysis of the domain and range for transformed rational functions, using the example of y=3/(1/(x-2)^2). The horizontal asymptote is determined to be y=3, and the vertical asymptote is found by setting the denominator to zero, resulting in x=2. The function is similar to 1/x^2, but it is reflected over the horizontal asymptote due to the negative sign, existing in the lower left and lower right regions. The domain is from negative infinity to 2 and from 2 to infinity, excluding x=2. The range is from negative infinity to 3, as the graph does not reach the horizontal asymptote. This paragraph reinforces the method of graphing rational functions using transformations and emphasizes the importance of identifying asymptotes, domain, and range.