Copyright © 2007 Pearson Education, Inc. Slide 4-2 Chapter 4: Rational, Power, and Root Functions...
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Transcript of Copyright © 2007 Pearson Education, Inc. Slide 4-2 Chapter 4: Rational, Power, and Root Functions...
Copyright © 2007 Pearson Education, Inc. Slide 4-2
Chapter 4: Rational, Power, and Root Functions
4.1 Rational Functions and Graphs
4.2 More on Graphs of Rational Functions
4.3 Rational Equations, Inequalities, Applications, and Models
4.4 Functions Defined by Powers and Roots
4.5 Equations, Inequalities, and Applications Involving Root Functions
Copyright © 2007 Pearson Education, Inc. Slide 4-3
4.1 Rational Functions and Graphs
• Rational function – quotient of two polynomials
p(x) and q(x), with q(x) 0.
• Examples
)()(
)(xqxp
xf
3521
)(,1
)(2
xx
xxf
xxf
Copyright © 2007 Pearson Education, Inc. Slide 4-4
• 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 © 2007 Pearson Education, Inc. Slide 4-5
4.1 The Reciprocal Function
Copyright © 2007 Pearson Education, Inc. Slide 4-6
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 © 2007 Pearson Education, Inc. Slide 4-7
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 © 2007 Pearson Education, Inc. Slide 4-8
4.1 The Rational Function f (x) = 1/x2
Copyright © 2007 Pearson Education, Inc. Slide 4-9
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.
Copyright © 2007 Pearson Education, Inc. Slide 4-10
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 © 2007 Pearson Education, Inc. Slide 4-11
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
