Rational Function Characteristics.notebook
November 17, 2016
Warm Up
Find the domain of each function.
1.
2.
3.
Homework Answers
factored form
1.
2.
VA
Holes
HA
x(2x+3)/(x-4)(x + 1)
all reals
except
x =4, -1
x = 4,
x = -1
NONE y = 2
1/(x+1)
all reals
except
x=-1
x = -1
NONE y = 0
(x-2)(x+2)/(x-2)
all reals
except
x=2
NONE
x=2
NONE
(x-1)(x+1)/(x-3)(x+1)
all reals
except
x=3, -1
x=3
x = -1
y=1
5/(x+5)(x-2)
all reals
except
x = -5, 2
x = -5
x=2
NONE y = 0
(x-7)/(x-7)(x-3)
all reals
except
x=7, 3
x=3
x=7
5/(x-3)
all reals
except
x =3
x=3
NONE y = 0
(x+5)(x-1)/2(x+5)(x-5)
all reals
except
x=-5, 5
x=5
x = -5
3.
4.
5.
6.
7.
8.
domain
y=0
y = 1/2
November 17, 2016
B. Asymptotes and Holes of Rational Functions continued
Pull
Rational Function Characteristics.notebook
Model examples as needed.
Examples: Find the vertical asymptotes and the x­values for holes in the graph of each function, if any.
Pull
2. Holes
If a factor of the denominator IS a factor of the numerator, then a hole in the graph occurs.
Hole: x = 0
Pull
1.
2.
Hole: none
Pull
3.
Hole: x = 1/2
Try these: Find the vertical asymptotes and holes, if any.
Drag each function to the shaded
box below to check your answer!
2.
1.
VA: x = 0 and x = 6
Hole: none
3.
VA: x = ­3
Hole: x = ­4
VA: x = 1
Hole: x = 0 and x = ­1
Rational Function Characteristics.notebook
November 17, 2016
B. Asymptotes and Holes of Rational Functions continued
3. Horizontal Asymptotes (HA)
To find the horizontal asymptote (HA) of a rational function, you MUST compare the degree of the numerator to the degree of the denominator.
(Only look at the term with the largest exponent in both the numerator and denominator.)
3 Cases: Let n = degree of numerator,
and let d = degree of denominator.
Examples: Find the horizontal asymptote, if any, of each function.
Pull
1. If n < d, then y = 0 is the HA.
2. If n = d, then y = a/b is the HA ("a" and "b" are the coefficients of the leading terms in the numerator and denominator)
3. If n > d, then there is NO HA.
Model all thre
Three practice
the next page
1.
HA: y = 0
2.
HA: y = 2/3
3.
HA: none
Rational Function Characteristics.notebook
November 17, 2016
Try these:
Find the horizontal asymptote, if any, of each function.
1.
y = 0
2.
y = ­1
3.
y = 1/2
Putting It ALL together!
Well.....almost all!
Find the domain, all asymptotes and holes.
1.
D:
VA:
Hole:
Click to check
your answers.
HA:
Rational Function Characteristics.notebook
November 17, 2016
Find the domain, all asymptotes and holes.
2.
D:
Click to check
your answers.
VA:
Hole:
HA:
C. Intercepts
1. Y-Intercept: substitute x = 0 into the equation.
a. If "y" exists, this is the value of they-intercept.
b. If "y" is undefined, there is NO y-intercept.
2. X-Intercepts (if any): set thenumerator equal to 0 and
solve for the x values.
Rational Function Characteristics.notebook
November 17, 2016
NOW!!!!
Find the domain, all asymptotes, holes and intercepts.
D:
3.
VA:
Click to check
your answers.
Hole:
HA:
x­intercept:
y­intercept:
Find the domain, all asymptotes, holes and intercepts.
4.
D:
VA:
Click to check
your answers.
Hole:
HA:
x­intercept:
y­intercept:
Homework: Worksheet Practice 5-2 Rational Functions
(see attachments)
Attachments
Practice 8­2 Rational Functions.doc