Sec 4 - University of MinnesotaSec 4.2 Rational Functions Graph f(x) = (x +3)2(x 4) Short Version: 1...
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Sec 4.2 Rational Functions Math 1051 - Precalculus I Rational Functions Sec 4.2
Transcript of Sec 4 - University of MinnesotaSec 4.2 Rational Functions Graph f(x) = (x +3)2(x 4) Short Version: 1...
Sec 4.2
Rational Functions
Math 1051 - Precalculus I
Rational Functions Sec 4.2
Sec 4.2 Rational Functions
Graph f (x) = (x + 3)2(x − 4)
Rational Functions Sec 4.2
Sec 4.2 Rational Functions
Graph f (x) = (x + 3)2(x − 4)Short Version:
1 Intercepts/zeros2 Even multiplicity touches. Odd multiplicity crosses.3 End behavior4 Turning points5 Behavior near zeros/intercepts6 Plot points
Rational Functions Sec 4.2
Sec 4.2 Rational Functions
Graph f (x) = (x + 3)2(x − 4)
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Rational Functions Sec 4.2
Graph f (x) = (x + 2)2(x − 2)2
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Rational Functions Sec 4.2
Graph f (x) = (x + 2)2(x − 2)2
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Rational Functions Sec 4.2
Rational functions
A rational function has the form
R(x) =p(x)q(x)
where p and q are polynomials and q 6= 0.
For example,
f (x) =x2 + 3x − 4
x − 2with x 6= 2.
Rational Functions Sec 4.2
Rational functions
A rational function has the form
R(x) =p(x)q(x)
where p and q are polynomials and q 6= 0.
For example,
f (x) =x2 + 3x − 4
x − 2with x 6= 2.
Rational Functions Sec 4.2
Rational functions
A rational function has the form
R(x) =p(x)q(x)
where p and q are polynomials and q 6= 0.
For example,
f (x) =x2 + 3x − 4
x − 2with x 6= 2.
Rational Functions Sec 4.2
Graphing Rational Functions
f (x) =1x2
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Rational Functions Sec 4.2
Graphing Rational Functions
f (x) =1x2
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Rational Functions Sec 4.2
Graphing Rational Functions
f (x) =1x2
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Rational Functions Sec 4.2
f (x) =−4
(x + 3)2 − 2
is a rational function
We can graph it using transformations...follow a particular point(maybe the point (1,1), for example)...
-6 -4 -2 2 4 6
-10
-8
-6
-4
-2
2
4
Rational Functions Sec 4.2
f (x) =−4
(x + 3)2 − 2
is a rational function
We can graph it using transformations...follow a particular point(maybe the point (1,1), for example)...
-6 -4 -2 2 4 6
-10
-8
-6
-4
-2
2
4
Rational Functions Sec 4.2
f (x) =−4
(x + 3)2 − 2
is a rational function
We can graph it using transformations...follow a particular point(maybe the point (1,1), for example)...
-6 -4 -2 2 4 6
-10
-8
-6
-4
-2
2
4
Rational Functions Sec 4.2
There are some important features to observe in the graph:
-6 -4 -2 2 4 6
-10
-8
-6
-4
-2
2
4
Rational Functions Sec 4.2
Vertical Asymptotes
Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.
The graph NEVER crosses a vertical asymptote.
To find vertical asymptotes:
Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)
Rational Functions Sec 4.2
Vertical Asymptotes
Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.
The graph NEVER crosses a vertical asymptote.
To find vertical asymptotes:
Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)
Rational Functions Sec 4.2
Vertical Asymptotes
Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.
The graph NEVER crosses a vertical asymptote.
To find vertical asymptotes:Reduce the fraction
Find x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)
Rational Functions Sec 4.2
Vertical Asymptotes
Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.
The graph NEVER crosses a vertical asymptote.
To find vertical asymptotes:Reduce the fractionFind x that make the denominator = 0
If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)
Rational Functions Sec 4.2
Vertical Asymptotes
Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.
The graph NEVER crosses a vertical asymptote.
To find vertical asymptotes:Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)
If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)
Rational Functions Sec 4.2
Vertical Asymptotes
Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.
The graph NEVER crosses a vertical asymptote.
To find vertical asymptotes:Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)
Rational Functions Sec 4.2
Find the vertical asymptotes of
f (x) =x2 + 3x + 2
x2 − 1
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
But there should be a hole at x = −1!
Rational Functions Sec 4.2
Find the vertical asymptotes of
f (x) =x2 + 3x + 2
x2 − 1
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
But there should be a hole at x = −1!
Rational Functions Sec 4.2
Find the vertical asymptotes of
f (x) =x2 + 3x + 2
x2 − 1
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
But there should be a hole at x = −1!Rational Functions Sec 4.2
Horizontal Asymptotes
Horizontal Asymptote: If, as x approaches +∞ or −∞, R(x)approaches some fixed number L, then the line y = L is ahorizontal asymptote.
For example, in the function f (x) = x2+3x+2x2−1 we just looked at
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
The line y = 1 is a horizontal asymptote.
Rational Functions Sec 4.2
Horizontal Asymptotes
Horizontal Asymptote: If, as x approaches +∞ or −∞, R(x)approaches some fixed number L, then the line y = L is ahorizontal asymptote.
For example, in the function f (x) = x2+3x+2x2−1 we just looked at
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
The line y = 1 is a horizontal asymptote.Rational Functions Sec 4.2
Two possibilities
Case 1: If (degree of top) < (degree of bottom), then y = 0 is ahorizontal asymptote.
f (x) =x + 2
x2 + 7x + 12
-6 -4 -2 2 4 6
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-5
5
10
Rational Functions Sec 4.2
Two possibilities
Case 1: If (degree of top) < (degree of bottom), then y = 0 is ahorizontal asymptote.
f (x) =x + 2
x2 + 7x + 12
-6 -4 -2 2 4 6
-10
-5
5
10
Rational Functions Sec 4.2
Two possibilities
Case 1: If (degree of top) < (degree of bottom), then y = 0 is ahorizontal asymptote.
f (x) =x + 2
x2 + 7x + 12
-6 -4 -2 2 4 6
-10
-5
5
10
Rational Functions Sec 4.2
Two possibilities
Case 2: If (degree of top) = (degree of bottom), then y = anbn
is ahorizontal asymptote.
f (x) =3x2 − 5x − 2
x2 − 1
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10
Rational Functions Sec 4.2
Two possibilities
Case 2: If (degree of top) = (degree of bottom), then y = anbn
is ahorizontal asymptote.
f (x) =3x2 − 5x − 2
x2 − 1
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-10
-5
5
10
Rational Functions Sec 4.2
Two possibilities
Case 2: If (degree of top) = (degree of bottom), then y = anbn
is ahorizontal asymptote.
f (x) =3x2 − 5x − 2
x2 − 1
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-5
5
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Rational Functions Sec 4.2
More possibilities
If (degree of top) = (degree of bottom) + 1
y = mx + b is called an oblique asymptote
f (x) =x2 + 3x − 4
x − 5
Rational Functions Sec 4.2
More possibilities
If (degree of top) = (degree of bottom) + 1
y = mx + b is called an oblique asymptote
f (x) =x2 + 3x − 4
x − 5
Rational Functions Sec 4.2
More possibilities
If (degree of top) = (degree of bottom) + 1
y = mx + b is called an oblique asymptote
f (x) =x2 + 3x − 4
x − 5
Rational Functions Sec 4.2
More possibilities
If (degree of top) = (degree of bottom) + 1
y = mx + b is called an oblique asymptote
f (x) =x2 + 3x − 4
x − 5
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Rational Functions Sec 4.2
More possibilities
If (degree of top) = (degree of bottom) + 1
y = mx + b is called an oblique asymptote
f (x) =x2 + 3x − 4
x − 5
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10
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Rational Functions Sec 4.2
More possibilities
What if (degree of top) > (degree of bottom) + 1 ?
There is no horizontal or oblique asymptote
f (x) =x4 − 50x + 2
x2 + 5
-6 -4 -2 2 4 6
-10
-5
5
10
15
20
Rational Functions Sec 4.2
More possibilities
What if (degree of top) > (degree of bottom) + 1 ?
There is no horizontal or oblique asymptote
f (x) =x4 − 50x + 2
x2 + 5
-6 -4 -2 2 4 6
-10
-5
5
10
15
20
Rational Functions Sec 4.2
More possibilities
What if (degree of top) > (degree of bottom) + 1 ?
There is no horizontal or oblique asymptote
f (x) =x4 − 50x + 2
x2 + 5
-6 -4 -2 2 4 6
-10
-5
5
10
15
20
Rational Functions Sec 4.2
More possibilities
What if (degree of top) > (degree of bottom) + 1 ?
There is no horizontal or oblique asymptote
f (x) =x4 − 50x + 2
x2 + 5
-6 -4 -2 2 4 6
-10
-5
5
10
15
20
Rational Functions Sec 4.2
One more example
f (x) =x3 − 1x − x2
-6 -4 -2 2 4 6
-6
-4
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2
4
6
With a hole at x = 1!
Rational Functions Sec 4.2
One more example
f (x) =x3 − 1x − x2
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
With a hole at x = 1!
Rational Functions Sec 4.2
Read section 4.3 for Friday.
We will learn techniques to graph rational functions by hand.
Rational Functions Sec 4.2
