Copyright © 2011 Pearson Education, Inc. Slide 4.1-1 4.1 Rational Functions and Graphs Rational...
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Copyright © 2011 Pearson Education, Inc. Slide 4.1-1 4.1 Rational Functions and Graphs Rational Function A function f of the form p/q defined by where p(x) and q(x) are polynomials, with q(x) 0, is called a rational function. Examples ) ( ) ( ) ( x q x p x f 3 5 2 1 ) ( , 1 ) ( 2 x x x x f x x f
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Transcript of Copyright © 2011 Pearson Education, Inc. Slide 4.1-1 4.1 Rational Functions and Graphs Rational...
Copyright © 2011 Pearson Education, Inc. Slide 4.1-1
4.1 Rational Functions and Graphs
Rational Function
A function f of the form p/q defined by
where p(x) and q(x) are polynomials, with q(x) 0, is
called a rational function.
Examples
)()(
)(xqxp
xf
3521
)(,1
)(2
xx
xxf
xxf
Copyright © 2011 Pearson Education, Inc. Slide 4.1-2
• The simplest rational function – the reciprocal function
4.1 The Reciprocal Function
xxf
1)(
.
theis 0 ,0 as )(
asymptotevertical
xxxf
.
theis 0,0, 1
asymptotehorizontal
yxx
Copyright © 2011 Pearson Education, Inc. Slide 4.1-3
4.1 The Reciprocal Function
Copyright © 2011 Pearson Education, Inc. Slide 4.1-4
4.1 Transformations of the Reciprocal Function
• The graph of can be shifted, translated, and reflected.
Example Graph
Solution The expression
can be written as
Stretch vertically by a
factor of 2 and reflect across
the y-axis (or x-axis).
xy
1
.2x
y
x2 .
12
x
xy
1
Copyright © 2011 Pearson Education, Inc. Slide 4.1-5
4.1 Graphing a Rational Function
Example Graph
Solution Rewrite y:
The graph is shifted left 1 unit and stretched
vertically by a factor of 2.
.1
2
x
y
11
21
2xx
y
xy
1
0:Asymptote Horizontal
1 :Asymptote Vertical
),1()1,( :Domain
y
x
Copyright © 2011 Pearson Education, Inc. Slide 4.1-6
4.1 Mode and Window Choices for Calculator Graphs
• Non-decimal vs. Decimal Window– A non-decimal window (or connected mode) connects
plotted points.
– A decimal window (or dot mode) plots points without connecting the dots.
• Use a decimal window when plotting rational functions such as
– If y is plotted using a non-decimal window, there would be a vertical line at x = –1, which is not part of the graph.
.1
2
x
y
Copyright © 2011 Pearson Education, Inc. Slide 4.1-7
4.1 Mode and Window Choices for Calculator Graphs
Illustration
Note: See Table for the y-value at x = –1: y1 = ERROR.
mode.dot and mode connectedin plotted 1
21 x
y
Copyright © 2011 Pearson Education, Inc. Slide 4.1-8
4.1 The Rational Function f (x) = 1/x2
Copyright © 2011 Pearson Education, Inc. Slide 4.1-9
4.1 Graphing a Rational Function
Example Graph
Solution
.1)2(
12
x
y
unit. 1down and
units 2left Shift
.1)2(
then ,1
)( If
12
2
x
xfyx
xf
Vertical Asymptote: x = –2; Horizontal Asymptote: y = –1.
