Sec37NotesRationalFunctions.notebook
October 10, 2012
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
y­values when ‘x’ is “close to 0”.
The next simplest rational function
The function is not defined at x=0. What happens to the y­values when ‘x’ is “close to 0”?
Sec37NotesRationalFunctions.notebook
October 10, 2012
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 Sec37NotesRationalFunctions.notebook
October 10, 2012
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.
Sec37NotesRationalFunctions.notebook
October 10, 2012
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.
Sec37NotesRationalFunctions.notebook
October 10, 2012
Graphing Rational Functions
• Factor the numerator and denominator.
• x­intercepts: the zeros of the numerator are the x­intercepts (unless they are also zeros of the denominator)
• y­intercepts: Plug in ‘x=0’ to find the y­intercept.
• 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.
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
End Behavior:
Zeros:
Vert Asymptotes:
Try typing any function into the following:
http://www.wolframalpha.com/
Sec37NotesRationalFunctions.notebook
October 10, 2012
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
End Behavior:
Zeros:
Vert Asymptotes:
Web Page to Graph Rational Functions
http://www.math.ohio­state.edu/~maharry/GeoGebra/GraphingRationalFunctions.html
http://www.math.ohio­state.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.
Sec37NotesRationalFunctions.notebook
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
End Behavior:
Zeros:
Vert Asymptotes:
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
End Behavior:
Zeros:
Vert Asymptotes:
October 10, 2012
Sec37NotesRationalFunctions.notebook
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
End Behavior:
Zeros:
Vert Asymptotes:
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
End Behavior:
Zeros:
Vert Asymptotes:
October 10, 2012
Sec37NotesRationalFunctions.notebook
October 10, 2012
Sketch the graph of the function. Think about zeros, asymptotes, multiplicities, and end behavior.
Vertical Asymptote:
Horizontal Asymptote: (End Behavior)
x­intercept:
y­intercept:
Find the inverse function of the following:
Then Graph it.
Sec37NotesRationalFunctions.notebook
October 10, 2012
Now graph the inverse function.
Vertical Asymptote:
Horizontal Asymptote: (End Behavior)
x­intercept:
y­intercept:
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.