Rational Functions and their graphs.....

Rational+Function+and+It's+Graph.pdf

1. RATIONAL FUNCTIONS AND
THEIR GRAPHS

2. Where p and q are polynomial functions and q is not the zero polynomial.
The domain consists of all real numbers except those for which the
denominator is 0.

3. Find the domain of the following rational functions:
All real numbers except -4 and 4.
All real numbers.

4. Finding Asymptotes
VERTICAL ASYMPTOTES()
43
52
2
2
−−
++
=
xx
xx
xR
Let’s set the bottom = 0 and
factor and solve to find where
the vertical asymptote(s) should
be.()()014 =+−xx
So there are vertical
asymptotes at x= 4 and x= -1.

5. If the degree of the numerator is less than
the degree of the denominator, the xaxis
is a horizontal asymptote. This is along
the liney= 0.
We compare the degrees of the polynomial in the numerator and the
polynomial in the denominator to tell us about horizontal asymptotes.()
43
52
2
+−
+
=
xx
x
xR
degree of bottom = 2
HORIZONTAL ASYMPTOTES
1
1 < 2

6. degree of bottom = 2
HORIZONTAL ASYMPTOTES
degree of top = 2
The leading coefficient is the
number in front of the highest
powered xterm.
12= ()
43
542
2
2
+−
++
=
xx
xx
xR 1
2
=y

7. ()
43
532
2
23
+−
+−+
=
xx
xxx
xR If the degree of the numerator is greater
than the degree of the denominator, then
there is not a horizontal asymptote, but an
oblique one. The equation is found by
doing long division and the quotient is the
equation of the oblique asymptoteignoring
the remainder.
OBLIQUE ASYMPTOTES -Slanted
degree of top = 3532
23
+−+ xxx 43
2
−−xx remainder a 5++x
Oblique asymptote at y= x
+ 5

8. SUMMARY OF HOW TO FIND ASYMPTOTES
To determine horizontal or oblique asymptotes, compare the degrees of the
numerator and denominator.
1.If the degree of the top < the bottom, horizontal asymptote along the x
axis (y= 0)
2.If the degree of the top = bottom, horizontal asymptote at y= leading
coefficient of top over leading coefficient of bottom
3.If the degree of the top > the bottom, oblique asymptote found by long
division.

9. STRATEGY FOR GRAPHING A RATIONAL FUNCTION

10. SKETCH THE GRAPH OF105
32
)(
+
−
=
x
x
xf

11. 105
32
)(
+
−
=
x
x
xf The vertical asymptote is x = -2
The horizontal asymptote is y = 2/5

12. 105
32
)(
+
−
=
x
x
xf -10-8-6-4-2 246810
-10
-8
-6
-4
-2
2
4
6
8
10

13. SKETCH THE
GRAPH OF:
g(x)=
1
x−1
Vertical asymptotes at??
Horizontal asymptote at??
y = 0

14. 
f(x)=
2
x Vertical asymptotes at??
x = 0
Horizontal asymptote at??
y = 0

15. 
h(x)=
−4
x Vertical asymptotes at??
x = 0
y = 0

16. SKETCH THE
GRAPH OF:
y=
1
x+3
−2
Vertical asymptotes at??x = 1
y = 0
Hopefully you remember,
y = 1/x graph and it’s asymptotes:
Vertical asymptote: x = 0
Horizontal asymptote: y = 0

17. We have the function:
y=
1
x+3
−2
But what if we simplified this and combined like terms:
y=
1
x+3
−
2(x+3)
x+3
y=
1−2x−6
x+3
y=
−2x−5
x+3
Now looking at this:
Vertical Asymptotes??
x = -3
Horizontal asymptotes??
y = -2

18. SKETCH THE
GRAPH OF:
h(x)=
x
2
+3x
x
x = 0
h(x)=
x(x+3)
x

19. FIND THE ASYMPTOTES OF EACH
FUNCTION:
y=
x
2
+3x−4
x 
y=
x
2
+3x−28
x
3
−11x
2
+28x 
y=
x
2
x
+
3x
x
−
4
x 
y=x+3−
4
x
Vertical Asymptote:
x = 0
y=
(x+7)(x−4)
x(x−7)(x−4)
Hole at x = 4
Vertical Asymptote:
x = 0 and x = 7
Horizontal Asymptote:
y = 0

20. WHAT MAKES A FUNCTION CONTINUOUS?
Continuous functions are predictable…
1) No breaksin the graph
A limit must exist at every x-value or the
graph will break.
2) No holesor jumps
The function cannot have undefined points
or vertical asymptotes.

21. Key Point:
Continuous functions can be
drawn with a single,
unbroken pencil stroke.

22. CONTINUITY OF POLYNOMIAL AND RATIONAL
FUNCTIONS

23. DISCONTINUITY
Discontinuity: a point at
which a function is not
continuous

24. DISCONTINUITY
Two Types of Discontinuities
1) Removable(hole in the graph)
2) Non-removable(break or vertical asymptote)
A discontinuity is calledremovableif a function can be made
continuous by defining (or redefining) a point.

26. DISCONTINUITY2
2
()
3 10
x
fx
xx
+
=
−−
Find the intervals on which these function are continuous. 2
( 2)( 5)
x
xx
+
=
+− 1
( 5)x
=
− 20x+= 2x=−
Vertical Asymptote:50x−= 5x=
Removable
discontinuity
Non-removable
discontinuity

27. DISCONTINUITY2
2
()
3 10
x
fx
xx
+
=
−− ( , 2) ( 2, 5) (5, )− −  −  

28. DISCONTINUITY2
2 , 2
()
4 1, 2
xx
fx
x x x
−
=
− +  2
lim( 2 )
x
x
−
→
− 2
2
lim( 4 1)
x
xx
+
→
−+ (2)f 4=− 3=− 4=− ( , 2] (2, )−  
Continuous on:

29. Determine the value(s) of xat which the function is
discontinuous. Describe the discontinuity as
removable or non-removable.2
2
1
()
56
x
fx
xx
−
=
−− 2
2
45
()
25
xx
fx
x
−−
=
− 2
2
10 9
()
81
xx
fx
x
++
=
− 2
2
4
()
28
x
fx
xx
−
=
−−
(A) (B)
(C)

30. DISCONTINUITY2
2
1
()
56
x
fx
xx
−
=
−−
(A)( 1)( 1)
( 6)( 1)
xx
xx
−+
=
−+ 1x=− 6x=
Non-removable discontinuity

31. DISCONTINUITY
(B)9x=− 9x=
Removable discontinuity( 9)( 1)
( 9)( 9)
xx
xx
++
=
+− 2
2
10 9
()
81
xx
fx
x
++
=
−

32. DISCONTINUITY
(C)5x= 5x=−
Non-removable discontinuity( 5)( 1)
( 5)( 5)
xx
xx
−+
=
−+ 2
2
45
()
25
xx
fx
x
−−
=
−

33. DISCONTINUITY
(D)2x=− 4x=
Removable discontinuity( 2)( 2)
( 4)( 2)
xx
xx
−+
=
−+ 2
2
4
()
28
x
fx
xx
−
=
−−

34. CONCLUSION
Continuousfunctions have no breaks, no holes, and no jumps.
If you can evaluate any limit on the function using only the substitution
method, then the function is continuous.