2.6 (Day One)Rational Functions & Their Graphs

• Objectives for 2.6– Find domain of rational functions.– Identify vertical asymptotes.– Identify horizontal asymptotes.– Use transformations to graph rational functions.– Graph rational functions.– Identify slant (oblique) asymptotes.

– Solve applied problems with rational functions.

Pg. 342 #2-8 (evens), 22-36(evens), 72a, 74a, 76a, 78a

Find the domain of each rational function.

1. f(x) =

2. f(x) =

3. f(x) =

2 25

5

x

x

2 25

x

x

2

5

25

x

x

Vertical asymptotes• A vertical asymptote is a vertical line that marks a single value of x excluded

from the domain of the function.

• Look for domain restrictions. If there are values of x which result in a zero denominator, these values would create either a hole in the graph or a vertical asymptote line.

• If the factor that creates a zero denominator cancels with a factor in the numerator, there is a hole.

• Example: Hole at x=-5

• If you cannot cancel the factor from the denominator, a vertical asymptote line exists.

• Example: Vertical line asymptotes at x=3 and -3

An asymptote is a line (or curve) that forms a boundary that a function approaches, moving closer and closer; sometimes over infinite regions.

2

4 4

9 ( 3)( 3)

x x

x x x

2 3 10 ( 2)( 5)2

5 5

x x x xx

x x

Find the vertical asymptote(s) or hole, if any, in the graph of each rational function.

4. 5.

6. 7.

2( )

1

xf x

x

2

1( )

1

xg x

x

2

1( )

1

xh x

x

2 4( )

2

xf x

x

What is the end behavior of this rational function? Meaning, what is the graph dong on the far left and far right?

• If you are interested in the end behavior, you are concerned with extremely large and extremely small values of x.

• As x approaches positive or negative infinity, the highest degree term becomes the only term of interest. The other terms become negligible (of very little importance) in comparison.

• So, only examine the ratio of the highest degree term in the numerator over the highest degree term of the denominator (ignore all others!)

• As x gets large, becomes

• THEREFORE, a horizontal asymptote exists, y=3

33

)( x

xxf

2

73)(

x

xxf

For the graph of

• If n<m (the numerator is of lesser degree than the denominator),

then the end behavior is limited by the horizontal asymptote

• If n=m (the numerator is of equal degree with the denominator),

then the end behavior limited by the horizontal asymptote

•If n (the numerator) is exactly one degree higher than m (the denominator),

then the end behavior is limited by a slanted line asymptote.

•If n (the numerator) is more than one degree higher than m (the denominator), then the end behavior is limited by an asymptote that can be a curve.

NOTE: The asymptotic curve is NOT covered in this precalculus textbook.

11 1 0

11 1 0

...( )

( ) ...

n nn n

m nm m

a x a x a x af xy

g x b x b x b x b

n

n

ay

b

0y

Find the horizontal or the slant asymptote, if any asymptote exists, of the graph of each of the following rational functions.

8.

9.

2

2

9( )

3 1

xf x

x

22 5 7( )

2

x xf x

x

2

9( )

3 1

xg x

x

10.

11.

3

2

9( )

3 1

xh x

x

22 5 7( )

2

x xf x

x

Graph of this rational function

62

238)(

2

x

xxxf

It has a vertical asymptote at x=3 and a slant asymptote at y = 4x+(21/2)