Warm-Up: January 12, 2012 Find all zeros of

1243 23 xxxxf

Homework Questions?

Rational Functionsand Their Graphs

Section 2.6

Objectives1. Find the domain of rational functions2. Use arrow notation3. Identify vertical asymptotes4. Identify horizontal asymptotes5. Graph rational functions6. Identify slant asymptotes7. Solve applied problems involving rational

functions

Rational Functions Rational Functions are quotients of

polynomial functions

The domain of a rational function is all real numbers except those that cause the denominator to equal 0

xqxp

xf

Example 1 (like HW #1-8) Find the domain of

64

82

x

xxf

You-Try #1 (like HW #1-8) Find the domain of

2

52

xx

xxf

Arrow Notation As x a+, f(x) ∞

“As x approaches a from the right, f(x) approaches infinity”

As x a-, f(x) -∞ “As x approaches a from

the left, f(x) approaches negative infinity”

As x ∞, f(x) 0 “As x approaches

infinity, f(x) approaches zero”

Vertical Asymptotes An asymptote is a line that the graph of f(x)

approaches, but does not touch. The line x=a is a vertical asymptote if f(x)

increases or decreases without bound as x approaches a. As x a+, f(x) ±∞ As x a-, f(x) ±∞

If “a” is a zero of q(x), but not a zero of p(x), then x=a is a vertical asymptote.

xqxp

xf

Example 2 (like HW #21-28) Find the vertical asymptotes, if any, of

xx

xxf

52

You-Try #2 (like HW #21-28) Find the vertical asymptotes, if any, of

16

42

x

xxf

Holes A hole is a point that is not part of the domain

of a function, but does not cause an asymptote.

If “a” is a zero of q(x), and a zero of p(x), then there is a hole at x=a

Holes generally are not distinguishable on a graphing calculator graph

Example of a Hole

2,22

222

42

xxxfx

xxxf

x

xxf

Horizontal Asymptotes The line y=b is a horizontal asymptote if

f(x) approaches “b” as x increases or decreases without bound As x ∞, f(x) b OR As x - ∞, f(x) b

Identifying Horizontal Asymptotes Only the highest degree term of the top and

bottom matter

Let “n” equal the degree of p(x), the numerator

Let “m” equal the degree of q(x), the denominator

If n<m, then the x-axis (y=0) is the horizontal asymptote

If n=m, then the line is the horizontal asymptote

If n>m, then f(x) does not have a horizontal asymptote

...

...

m

m

nn

xb

xa

xq

xpxf

m

n

b

ay

Example 3 (like HW #29-33) Find the horizontal asymptote, if any, of each

function

23

6

23

6

23

62

3

2

2

2

x

xxh

x

xxg

x

xxf

Warm-Up: January 17, 2012 Find the horizontal asymptotes, if any, of:

Find the vertical asymptotes, if any, of

34

23

23

3

2

3

7

23

7

23

76

23

xx

xxxh

xx

xxxg

xx

xxf

65

232

2

xx

xxxf

Homework Questions?

You-Try #3 (like HW #29-33) Find the horizontal asymptote, if any, of each

function

7

23

7

23

7

234

3

3

3

2

3

x

xxxh

x

xxxg

x

xxxf

Graphing Rational Functions1. Find the zeros of p(x), the numerator2. Find the zeros of q(x), the denominator3. Identify any vertical asymptotes (numbers that are

zeros of q(x) but not zeros of p(x)). Draw a dashed line.

4. Identify any holes (x-values are numbers that are zeros of both p(x) and q(x))

5. Identify any horizontal asymptotes by examining the leading terms. Draw a dashed line.

6. Find f(-x) to determine if the graph of f(x) has symmetry:

If f(-x)=f(x), then there is y-axis symmetry If f(-x)=-f(x), then there is origin symmetry

xqxp

xf

Graphing Rational Functions, cont.7. Find the y-intercept by evaluating f(0)8. Identify the x-intercepts (numbers that are

zeros of p(x) but not q(x))9. Pick a few more points to plot10. Draw a curve through the points,

approaching but not touching the asymptotes. If there was a hole identified in step 4, put an open circle at that x-value.

11. Check your graph with a graphing calculator. Remember that it does not properly display asymptotes and holes.

Example 4 (like HW #37-58) Graph

62

2

xx

xxf

You-Try #4 (like HW #37-58) Graph

1

3

x

xxf

Warm-Up: January 18, 2012 Determine any and all asymptotes and holes

of:

3

2

3

62

x

xxxg

You-Try #5 (like HW #37-58) Graph

1

42

2

x

xxf

Slant Asymptotes A slant asymptote is a line of the form

y=mx+b that the graph of a function approaches as x±∞

The graph of f(x) has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator

Find the equation of the slant asymptote by division (synthetic or long), and ignore the remainder

xqremainder

bmxxq

xpxf

Example 7 (like HW #59-66) Find the slant

asymptote and graph

1

12

x

xxxf

You-Try #7 (like HW #59-66) Find the slant

asymptote and graph

x

xxf

42

Warm-Up: January 19, 2012 Determine any and all asymptotes and holes

of:

1

452

x

xxxf

16

452

2

x

xxxg

Applications of Rational Functions The average cost of producing an item

Chemical concentrations over time

Used in numerous science and engineering fields to approximate or model complex equations

n

cnCnC fixed

2

5( )

0.01 3.3

tC t

t

Example 8 (page 322 #70)The rational function

describes the cost, C(x), in millions of dollars, to inoculate x% of the population against a particular strain of the flu.

a) Find and interpret C(20), C(40), C(60), C(80), and C(90)

b) What is the equation of the vertical asymptote? What does this mean in terms of the variables of the function?

c) Graph the function

x

xxC

100

130

Assignment Page 321 #1-39 odd, 59, 67