Transcript Chapter 2

Definition of a Rational Function
Any function of the form
N(x)
f(x) 
D(x)
Where N(x) and D(x) are polynomials and
D(x) is not the zero polynomial
Examples.
3x  5
f(x) 
x4
3x  5
f(x) 
7
4x 2  3x  5
f(x) 
5x  1
not a rational function
f ( x) 
2x  1
x 1
x=1 is a vertical
asymptote because _____
as x  1 , y  
and
as x  1 ,
Vertical
asymptote
at x = 1
y  
f ( x) 
1
 x  1
2
x=1 is a vertical
asymptote because_____
as x  1 , y  
and
as x  1 ,
Vertical
asymptote
at x = 1
y
f (x ) 
Given: f ( x ) 
x5

x 2  25
N (x )
D (x )
1
x5

 x  5  x  5  x  5
vertical asymptote ( set denominator = 0 of reduced fraction)
x+5=0
x = -5
There is a vertical asymptote at x = -5
There is a hole at x = 5 ( the zeros of common factors)
Graph of
f ( x) 
x5
x 2  25
Find all vertical asymptotes and holes of f ( x ) 
there are no common factors
2x  1
x2
no holes
x = 2 is a vertical asymptote because 2 is a zero of the denominator in
the reduced form.
Find all vertical asymptotes and holes of
f ( x) 
3x
x2  2x  3
Find all vertical asymptotes and holes of
f ( x) 
2 x  14
x2  6x  7
f ( x) 
2x  1
x 1
y=2 is a horizontal
asymptote because___
as x   , y  2
and
as x   , y  2
horizontal
asymptote
at y = 2
Horizontal Asymptotes
f ( x) 
1
 x  1
2
y=0 is a horizontal
asymptote because ____
as x   , y  0
and
as x   , y  0
horizontal
asymptote
at y = 0
(let n = degree of numerator
a.
and
d = degree of denominator )
If n < d, then y = 0 is the horizontal asymptote
Examples:
y
x5
3x2  5 x  1
y
2x  3
x5  5 x3  1
Horizontal asymptote at y = 0
Horizontal asymptote at y = 0
an
b. If n = d, then y 
bn
Leading coefficient
of numerator
is the horizontal asymptote
Leading coefficient
of denominator
Examples:
3x  5
y
2x  6
10 x 2  5 x  5
y
5x2  2x  6
Horizontal asymptote at y = 3/2
Horizontal asymptote at y = 10/5 = 2
c. If n > d, then there is no horizontal asymptote
x2  x  4
y
x2
3 x4  3 x 3  4 x  1
y
x2  2
No horizontal asymptotes
No horizontal asymptotes
Identify any vertical or horizontal asymptotes, and any
holes in the graph
x5
1. y 
x3
x1
2. y  2
x 1
x2  x  2
3. y 
x1
2x 2  x  6
4. y 
2x  3
3x2
5. y  2
x 1
6. y 
x1
x 2  2x  3
Slant Asymptotes occur when the degree of the numerator
is exactly one more than the degree of the denominator of the
reduced fraction .
To find the equation of a slant asymptote use long division.
Ex. Find the equation of the slant asymptote of the equation
x3  x2  x  1
y
x2  x  1
y  x
2 x  1
x2  x  1
Equation of the horizontal asymptote is
y x
x3  x2  x  1
y
x2  x  1
Find the equation of the slant asymptote
x2  x
y
x 1
Equation of the horizontal asymptote is y = x-2
Find the equation of the slant asymptote
x2  x  2
y
x 1
Equation of the horizontal asymptote is y = x
State the domain of the function
Find and plot the y-intercept by evaluating f(0)
Find and plot the x-intercepts by finding the zeros of the numerator
Sketch the vertical asymptotes using dashed vertical lines, and holes
using open circles.
Find and sketch any horizontal asymptote using dashed lines.
Find and sketch any slant asymptote using dashed lines.
Plot at least one point between and one point beyond each x-intercept
and vertical asymptote.
Use smooth curves to complete the graph
Average Cost
The cost c of producing x units of a product is given by
C  0.2x2  10x  5
And the average cost per unit is given by
C
0.2x 2  10x  5
C 

, x  0
x
x
Graph the average cost function, and estimate the number of units
that should be produced to minimize the average cost per unit.
Medicine.
The concentration C of a chemical in the bloodstream t hours after
injection into muscle tissue is given by
3t2  t
C  3
, t  0
t  50
a. Determine the horizontal asymptote of the function and interpret
its meaning in the context of the problem
b. Graph the function and approximate the time when the
bloodstream concentration is the greatest.
c. When is the concentration less than 0.345?