Objectives Graph a rational function, find its
domain, write equations for its asymptotes, and identify any holes (point discontinuity) in its graph.
Glossary Termsasymptote
horizontal asymptote
point discontinuity
rational function
vertical asymptote
y-intercept
Rational Function
An equation in the form
Where p(x) and q(x) are polynomial functions and q(x)0.
p x
f xq x
When graphing rational functions
State domainFind Vertical Asymptote(s)Find Point of Discontinuity in the graph
(HOLES) Find Horizontal Asymptote Find the y-intercepts & x-intercepts Sketch
3f x
x 1 x 5
State DomainFind Vertical Asymptote(s)Find Point of Discontinuity in the graph
Domain: (-∞,-5) U (-5,1) U (1,∞)Vertical Asymptote(s) x=1 & x=-5Find Point of Discontinuity in the graph - none
3f x
x 1 x 5
Find Horizontal Asymptote
Compare the degree of the numerator to the degree of the denominator
degree of the numerator < degree of the denominator
H.A. y = 0
3f x
x 1 x 5
Find Horizontal Asymptote
Compare the degree of the numerator to the degree of the denominator
degree of the numerator < degree of the denominator
H.A. y = 0
degree of the numerator = degree of the denominator
H.A. y = the ratio of the lead coefficients.
degree of the numerator > degree of the denominator
none
3f x
x 1 x 5
Find y-intercept
Substitute zero for x and find the value of the function
3
f 00 1 0 5
3f 0
5
Sketch the graphVertical Asymptotes
Horizontal Asymptote
y-intercept
3f x
x 1 x 5
Plot a few points
x y
-10 .05
-5.5 .92
-4.5 -1.1
.5 -1.1
1.5 .92
10 .02
2
2
x 36f x
x 7x 6
State the domainFind Vertical Asymptote(s)Find Point of Discontinuity in the graph
Domain: (-∞,-6) U (-6,-1) U (-1,∞)Vertical Asymptote(s) x=-1Find Point of Discontinuity at -6
4
x 6 x 6
f xx 6 x 1
Find Horizontal Asymptote
Compare the degree of the numerator to the degree of the denominator
2
2
x 36f x
x 7x 6
degree of the numerator = degree of the denominator
H.A. y = the ratio of the lead coefficients.
y = 1
Find y-intercept Substitute zero for x and find the value of
the function
2
0 36f 0
0 7 0 6
36f 0
6
2
2
x 36f x
x 7x 6
f 0 6
Sketch the graphVertical Asymptote
Point of Discontinuity
Horizontal Asymptote
y-intercept
Plot a few points
x y
-10 1.8
-2 8
-.5 -13
1 -2.5
10 .4
2
2
x 36f x
x 7x 6
2x 16
f xx 4
State the Domain
Find Vertical Asymptote(s)Find Point of Discontinuity in the graph
Domain: (-∞,-4) U (-4,∞)Vertical Asymptote(s) NONEFind Point of Discontinuity at -4
6
x 4 x 4f x
x 4
Find Horizontal Asymptote
Compare the degree of the numerator to the degree of the denominator
degree of the numerator > degree of the denominator
H.A. there is no horizontal asymptote
2x 16
f xx 4
Find y-intercept Substitute zero for x and find the value of
the function
0 16f 0
0 4
16f 0
4
2x 16
f xx 4
f 0 4
Sketch the graphPoint of Discontinuity
y-intercept
Plot a few points
x y
-5 -9
-3 -7
4 0
2x 16
f xx 4
Homework – day 1
Rational Functions Worksheet 1, part A & B
Graphing Rational Function Worksheet #1-3, 5
Graphing Rational ExpressionsOblique Asymptotes
To find Oblique (Slant) Asymptotes you will need to divide.
2 9 20
3x x
f xx
3 2
2
3 2 3 41
x x xg x
x
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