ERF - Graphing Rational Functions Lesson

Graphing Rational Functions

A rational function is a function of the form equation image indicator $LaTeX: f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ , where $LaTeX: p\left(x\right)\:and\:q\left(x\right)$ $p\left(x\right)\:and\:q\left(x\right)$ are polynomials and $LaTeX: q\left(x\right)\ne0$ $q\left(x\right)\ne0$ . A hyperbola is the graph of a rational function of the form equation image indicator $LaTeX: f\left(x\right)=\frac{a}{x-h}+k$ $f\left(x\right)=\frac{a}{x-h}+k$ , whose center is (h, k), and asymptotes are x = h and y = k. Rational functions of the form $LaTeX: y=\frac{ax+b}{cx+d}$ $y=\frac{ax+b}{cx+d}$ also have graphs that are hyperbolas. The vertical asymptote occurs at the x - value that makes the denominator zero, and the horizontal asymptote is the line equation image indicator $LaTeX: y=\frac{a}{c}$ $y=\frac{a}{c}$ .

When investigating the graphs of rational functions, it is important to find various aspects of these graphs, such as domain, range, roots or zeros, the end behavior, the vertical asymptotes, the horizontal asymptotes, intervals of increase, intervals of decrease, and any points along the graph where there is(are) a "hole(s)." "Holes," vertical asymptotes, and horizontal asymptotes are very important features because they show parts of rational function graphs that are discontinuous.

It is important to state some information regarding horizontal asymptotes.

Given the following rational function, equation image indicator $LaTeX: f\left(x\right)=\frac{ax^n+...\:\longleftarrow nth\:degree\:of\:the\:polynomial}{bx^m+...\longleftarrow nth\:degree\:of\:the\:polynomial}$ $f\left(x\right)=\frac{ax^n+...\:\longleftarrow nth\:degree\:of\:the\:polynomial}{bx^m+...\longleftarrow nth\:degree\:of\:the\:polynomial}$

There are three scenarios for finding the Horizontal Asymptotes, and they are the following:

If n<m, then the x - axis is the horizontal asymptote.
If n = m, then the horizontal asymptote is the line $LaTeX: y=\frac{a}{b}$ $y=\frac{a}{b}$ .
If n >m, then there is No horizontal asymptote. There is a "slant diagonal" or "oblique asymptote." In order to find the oblique asymptote, use long division to divide the numerator by the denominator. The quotient is the "oblique asymptote

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Meticulously watch and take notes on the following videos that discuss and model the concepts involved in graphing rational functions. As you watch the teaching videos, pay close attention to why discontinuities are so important. After watching these teaching videos, complete the practice items provided so that you will master these concepts.

Graphing Rational Functions Practice

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