a rational function is a function of the form: where p and q are polynomials
TRANSCRIPT
RATIONALFUNCTIONS
A rational function is a function of the form:
xqxp
xR where p and q are polynomials
What’s domain?
The domain of a function is the set of all possible “input” values.
xqxp
xR We need to make sure the denominator 0
What can we NEVER divide by?!
What’s the domain?
x
xxR
3
5 2
What’s the domain?
22
3
xx
xxH
What’s the domain?
45
12
xx
xxF
Find the domain of
Now, graph it!
1
x
xxf
verticalasymptote!
Asymptote
An asymptote is a line that a curve approaches, but never touches!
Find the domain of the following functions. Then graph them to find their vertical asymptotes.
3
4.
2
x
xxga
)4)(2(
3.
2
xx
xxhb
What do you notice about
the domain of these
functions and their
asymptotes?
Recap: How do we determine vertical asymptotes?
Let’s take a look at the most recent example. Notice any other asymptotes?
horizontalasymptote!
)4)(2(
3.
2
xx
xxhb
Horizontal Asymptotes
To find a horizontal asymptotes, we focus on the degree of the numerator and the denominator.
3
2
)5(
)4()4)(3(
x
xxx What’s the degree?
What’s the degree?
How do we use degrees to find the horizontal asymptote?
BOBO BOTN EATS DC
BOBOBigger On Bottom, y = O
23
2
)2()5(
)4)(3(
xx
xx Degree of 3
Degree of 5
The degree is bigger on the bottom, so the horizontal asymptote is the line y = 0.
BOTNBigger On Top, No HA
2
23
)2)(5(
)2()4)(3(
xx
xxx Degree of 6
Degree of 3
The degree is bigger on the top, so there is no horizontal asymptote.
EATS DCExponents Are The Same, Divide Coefficients
2
2
)2)(5(3
)2()4(6
xx
xx Degree of 3
Degree of 3
The degrees are the same, so divide the leading coefficients. The horizontal asymptote is y = 2.
Find the horizontal asymptotes!
22
32
)2()5(7
)1()2()4(5.1
xx
xxx
2
2
)1)(5(4
)2()3(5.2
xx
xx
23
2
)2()5(
)2()4(2.3
xx
xx
Quick Recap!
Vertical AsymptoteUse the denominator
of the fraction.
Determine the values that make the denominator = 0.
Horizontal AsymptoteFind the degree of the
numerator and denominator.
Use BOBO BOTN EATS DC to find the horizontal asymptote.