1. No category

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 4-2
4.1 Rational Functions and Graphs
• Rational function – quotient of two polynomials
p(x) and q(x), with q(x)  0.
p( x)
f ( x) 
q( x)
• Examples
1
f ( x)  ,
x
Copyright © 2007 Pearson Education, Inc.
x 1
f ( x) 
2 x  5x  3
2
Slide 4-3
4.1 The Reciprocal Function
•
The simplest rational function – the reciprocal
function
1
f ( x) 
x
f ( x)   as x  0, x  0 is the
vertical asymptote.
Copyright © 2007 Pearson Education, Inc.
x  , 1x  0, y  0 is the
horizontal asymptote.
Slide 4-4
4.1 The Reciprocal Function
Copyright © 2007 Pearson Education, Inc.
Slide 4-5
4.1 Transformations of the Reciprocal
Function
The graph of y  1 can be shifted, translated, and
x
reflected.
2
Example Graph y   .
x
Solution
The expression
2
1
 can be written as  2 .
x
 x
1
Stretch y  vertically by a
x
factor of 2 and reflect across
the y-axis (or x-axis).
•
Copyright © 2007 Pearson Education, Inc.
Slide 4-6
4.1 Graphing a Rational Function
Example
Solution
2
.
Graph y 
x 1
2
1 

 2
Rewrite y: y 

x  1  x  1
y
1
x
The graph
is shifted left 1 unit and stretched
vertically by a factor of 2.
Domain : (,1)  (1, )
Vertical Asymptote : x  1
Horizontal Asymptote : y  0
Copyright © 2007 Pearson Education, Inc.
Slide 4-7
4.1 The Rational Function f (x) = 1/x2
Copyright © 2007 Pearson Education, Inc.
Slide 4-8
4.1 Graphing a Rational Function
1
 1.
Graph y 
( x  2)
Example
2
Solution
1
If f ( x)  , then
x
y  f ( x  2 )  1.
2
Shift
1
x2
left 2 units
and down 1 unit.
Vertical Asymptote: x = –2;
Horizontal Asymptote: y = –1.
Copyright © 2007 Pearson Education, Inc.
Slide 4-9
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
2
.
functions such as y 
x 1
– 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.
Copyright © 2007 Pearson Education, Inc.
Slide 4-10
4.1 Mode and Window Choices for
Calculator Graphs
Illustration
2
y1 
plotted in connected mode and dot mode.
x 1
Note: See Table for the y-value at x = –1: y1 = ERROR.
Copyright © 2007 Pearson Education, Inc.
Slide 4-11