chapter 4: polynomial & rational functions 4.4: rational functions essential question: how can...
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Chapter 4: Polynomial & Rational Functions4.4: Rational Functions
Essential Question: How can you determine the vertical and horizontal asymptotes of an equation?
4.4: Rational Functions Domain of Rational Functions
The domain is the set of all real numbers that are not zeros of its denominator.
Example 1: The Domain of a Rational Function Find the domain of each rational function
2
1( )f x
x
2
2
3 1( )
6
x xg x
x x
All real numbers except x = 0
All real numbers except x2 – x – 6 = 0(x – 3)(x + 2) = 0, so all real numbers except for 3 and -2
4.4: Rational Functions Properties of Rational Graphs
Intercepts As with any graph, the y-intercept is at f(0) The x-intercepts are when the numerator = 0 and the
denominator does not equal 0.
Example Find the intercepts of
y-intercept:
x-intercepts: Neither are solutions of x – 1 = 0
so both are x-intercepts
2 2( )
1
x xf x
x
20 0 2 2
(0) 20 1 1
f
2 2 0
( 2)( 1) 0
2 1
x x
x x
x x
4.4: Rational Functions Properties of Rational Graphs (continued)
Vertical Asymptotes Whenever only the denominator = 0 (numerator is not
0) Vertical asymptotes will either spike up to ∞ or down to
-∞
Big-Little Concept Dividing by a small number results in a large
number Dividing by a large number results in a small
number.
1little
big 1
biglittle
4.4: Rational Functions Behavior near a Vertical Asymptote
Describe the graph of near x = 2
As x gets closer and closer to 2 from the right (2.1, 2.01, 2.001, …) the denominator becomes a really small positive number. Division by a small positive number means the graph
of f(x) approaches ∞ from the rightAs x gets closer and closer to 2 from the left
(1.9, 1.99, 1.999, …) the denominator becomes a really small negative number Division by a small negative number means the graph
of f(x) approaches -∞ from the left
1( )
2 4
xf x
x
4.4: Rational Functions Holes
When a number c is a zero of both the numerator and denominator of a rational function, the function might have a vertical asymptote, or it might have a hole.
Example #1
But this is not the same as the function g(x) = x + 2, as f(2) = while g(2) = 4, so though they may look the same, f(x) has a hole at x = 2
Example #2
The graph of x2/x3 looks the same as 1/x, and has a vertical asymptote, as neither of the functions are defined at x = 0
2 ( 2)
2
4 ( 2)( ) 2
2
x xf x x
x
x
x
0
0
2
3
1( )
xf x
x x
4.4: Rational Functions Holes
If and a number d exists such that g(d) and h(d) = 0 If the degree of the numerator is greater than (or equal
to) the degree of the denominator after simplification, then the function has a hole at x = d = hole @ x = 5
If the degree of the denominator is greater than the degree of the numerator after simplification, then the function has a vertical asymptote at x = d = asymptote @ x = 5
It is far easier to first determine the domain of the function, and then visually inspect to see whether “hiccups” in the domain are holes or asymptotes.
( )( )
( )
g xf x
h x
2( 5)
5
x
x
2
5
( 5)
x
x
4.4: Rational Functions End Behavior (Horizontal Asymptotes)
The horizontal asymptote is found by determining what the function will be when x is extraordinarily large
When x is large, a polynomial function behaves like its highest degree term
Example #1 List the vertical asymptotes and describe the end
behavior.
There is a vertical asymptote at x = 5/2
Both numerator and denominator have the same degree, so the horizontal asymptote is at y = -3/2
3 6( )
5 2
xf x
x
3 6 3 6 3 3( )
5 2 2 5 2 2
x x xf x
x x x
4.4: Rational Functions Asymptotes, Example #2
Vertical asymptotes at: x2 – 4 = 0 (x – 2)(x + 2) = 0 Vertical asymptotes at x = 2 or x = -2
Horizontal Asymptotes at: As x becomes large, 1/x becomes small. So horizontal asymptote at y = 0.
2( )
4
xf x
x
2 2
1( )
4
x xf x
x x x
4.4: Rational Functions Asymptotes, Example #3
Vertical asymptotes at: x3 + 1 = 0 Graphing tells you there is only one root, at x = -1 Vertical asymptotes at x = -1
Horizontal Asymptotes at:
Horizontal asymptote at y = 2.
3
3
2( )
1
x xf x
x
3 3
3 3
2 2 2( )
1 1
x x xf x
x x
4.4: Rational Functions Assignment
Page 290 1 – 49, odd problems
Due tomorrow Show work
Only worry about holes, vertical and horizontal asymptotes. Disregard stuff on slant/parabolic asymptotes Ignore the graphing (you have a graphing calculator for
that)