Download Math 140 Lecture 10 y = 2x-6 y = 2x3-8x2

Graph. On the graph mark the x and y-intercepts. Mark
the vertical and horizontal asymptotes with their
Rational functions and their graphs
equations ( y = a or x = a ) .
DEFINTION. A rational function is a ratio of two
a
is
a key number iff fa  0 or f(a )  undefined.
polynomials. It is reduced if the top and bottom have no
Key intervals lie before, between and after key numbers.
common factors.
Like polynomials, rational functions have smooth graphs. In each key interval, calculate a key value.
2x6
`y  2x 3 8x 2 .
But they may have asymptotes.
Math 140
Lecture 10
`In the graphs below, x = 0 (the y-axis) is the vertical
asymptote, y = 0 (the x-axis) is the horizontal asymptote.
Let v.a. and h.a. abbreviate vertical and horizontal asymptotes.
y=1/x2
y=1/x
y=1/x3
y=0
y=0
x=0
y=0
h.a.
lead term:
x=0
y= -1/x
y= -1/x2
x=0
y=0
y= -1/x
x=0
v.a.
3
y=0
2x
2x 3

1
x2
horizontal asymptote:
y =0
key values: f (-1) = 4/5, f (1) = 2/3, f (7/2) = -4/49, f (5) = 2/25
x=0
v For odd degree vertical asymptotes, one side goes to
+5, the other to -5.
See 1/x and -1/x.
v For even degree vertical asymptotes, both sides go to
See 1/x2 and -1/x2.
+5 or both go to -5.
x=0
v.a.
0
y=0
h.a.
4
3
x=4
v.a.
DEFINITION. For rational functions, the leading term is the
reduced ratio of the leading terms of the top and
bottom.
Recall, to get the leading term of a factored polynomial, replace
each factor to its leading term and then simplify.
`Rational functions:
Leading terms:
Hor. asymptote:
x3
vertical asymptotes:
x = 0 (deg 2)
x = 4 (deg 1)
x=0
v.a.
y=0
h.a.
x3
2x3
reduce and factor: = 2x 3 4x 2  = x 3 4x 2 = x 2 x4
y-intercept: none
x-intercept: 3 (deg 1)
x
1x
1
y  1
x 2 1
3x 2
1
3
y  13
x15x2
x3 2 12x 3
5
8x 3
y0
12x 3
2x6
2
4x
none
`y 
2x 3 8x 2
2x6 .
x 2 x4
reduce and factor: x3
y-intercept: 0
x-intercepts: 0 (deg 2)
4 (deg 1)
vert. asymp.: x = 3 (deg 1)
For a reduced rational function:
v x-intercepts (roots) occur where the top is 0.
If the root has degree n, the x-intercept looks like that of y = xn or y
= -xn.
3
2x
2
lead term:
2x  x
hor. asymp.: none
key values: f (-1) = 5/4, f (1) = 3/2, f (7/2) = -49/4, f (5) = 25/2
v If the bottom is 0 at a, then x=a is a vertical asymptote.
If the factor has degree n, the vertical asymptote looks like that of
y =1/xn or y = -1/xn.
v As xƒ +5, the graph resembles the graph of the
leading term which is either a constant b, a fraction
a/xn or a term axn of positive degree.
(1) If a constant b, then y = b is a horizontal asymptote.
(2) If it is a/xn, then y = 0 is a horizontal asymptote.
(3) If it is axn, there is no horizontal asymptote.
x=3
v.a.
0
3
4