Sec. 3.7 Rational Functions and their Graphs. A rational function is of the form: where P(x) and Q(x) are Polynomials

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1 Sec. 3.7 Rational Functions and their Graphs A rational function is of the form: where P(x) and Q(x) are Polynomials The Domain of r(x) is all values of x where Q (x) is not equal to zero. The simplest rational function The function is not defined at x=0. What happens to the y values when x is close to 0. The next simplest rational function The function is not defined at x=0. What happens to the y values when x is close to 0?

2 What happens to a graph near a zero in the denominator? Vertical Asymptote y approaches negative infinity as x approaches 0 from the left is written as as y approaches positive infinity as x approaches 0 from the right is written as as What happens to a graph when x get large (positive or negative)? End Behavior y approaches zero as x goes to positive infinity is written as as y approaches zero as x goes to negative infinity is written as as

3 Asymptotes : a line (vertical, horizontal or slanted) that the graph gets closer and closer to as one travels along the line. A vertical line (x=a) is a vertical asymptote of y=f(x) if y approaches as x approaches a from the left or the right. A horizontal line (y=b) is a horizontal asymptote of y=f(x) if y approaches b as x approaches a positive or negative infinity. Transformations of This function can be graphed by synthetic division and transformations as in Example 2 in the book, but I think it can be done much easier by the following methods, which also work for more complicated rational functions.

4 Vertical asymptotes of Rational Functions These occur when the denominator is equal to zero. (Unless the numerator is also equal to zero at that point). What happens to the graph when x=3 as x approaches 3 from the left? Set x=2.99 From the right? Set x=3.01 End Behavior and Horizontal Asymptotes of Rational Functions Similar to the end behavior of a polynomial being determined by the leading term, the end behavior of a rational function is determined by the leading terms of the numerator and of the denominator.

5 Graphing Rational Functions Factor the numerator and denominator. x intercepts: the zeros of the numerator are the x intercepts (unless they are also zeros of the denominator) y intercepts: Plug in x=0 to find the y intercept. Vertical asymptotes: These occur at the places where the denominator is equal to zero. Behavior of the graph near vertical asymptotes. The book suggests using test values to determine if the graph goes to positive or negative infinity on each side of the asymptote. I believe it is easier to determine this by looking at the multiplicities of the factors in the numerator and denominator. We ll see examples soon. Horizontal Asymptotes: Find the function that describes the end behavior of the function by considering only the leading terms of the numerator and denominator. Draw the intercepts and asymptotes of the graph on the axes. Try typing any function into the following:

6 Web Page to Graph Rational Functions state.edu/~maharry/geogebra/graphingrationalfunctions.html state.edu/ ~maharry/geogebra/graphingrationalfunctionswithmultiplicities.html Graphing Rational Functions Start drawing the graph following the end behavior of the graph. If there is a horizontal asymptote at y=0, use a test value to determine if the graph is small but positive or small bu negative. Draw the graph from left to right following the rules when you approach a zero or an asymptote. If the multiplicity is odd, then the sign of the function changes when you pass through a zero or switch sides of an asymptote. If the multiplicity is even, then the sign of the function stays the same when you pass through a zero or switch sides of an asymptote.

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9 Vertical Asymptote: Horizontal Asymptote: (End Behavior) x intercept: y intercept: Find the inverse function of the following: Then Graph it.

10 Now graph the inverse function. Vertical Asymptote: Horizontal Asymptote: (End Behavior) x intercept: y intercept: Problem: The speed of a canoe in still water is 6 mph. Suppose the speed of the current is "x" mph. Find a formula for the time it takes to paddle 3 miles up stream, turn around and paddle back to the starting point.