2.6 & 2.7 Rational Functions 2.6 & 2.7 Rational Functions and Their Graphsand Their Graphs

2.6 & 2.7 Rational Functions 2.6 & 2.7 Rational Functions and Their Graphsand Their Graphs

Objectives: •Identify and evaluate rational functions•Graph a rational function, find its domain and range, write equations for its asymptotes, identify any holes in its graph, and identify the x- and y- intercepts

What is a Rational Expression?

• A rational expression is the quotient of two polynomials.

• A rational function is a function defined by a rational expression.

2

3

5

3

( )( 4)( 4)

( 3)( )

27

xy

xx

g xx x

xf x

x

3

1

2

7

( )2 1

( )5

x

x

yx

xg x

x

xf x

Simplify

2

2

x 7x 18x 8x 9

2

2

x 7x 18x 8x 9

(x 9)(x 2)(x 9)(x 1)

x 2x 1

Find the Domain

Find the domain of 2

2

4 21( )

9 36

x xh x

x x

To find the domain of a rational function, you 1st must find the values of x for which the denominator equals 0. x2 – 9x – 36 = 0

(x – 12)(x + 3) = 0 x = 12 or -

3

The domain is all real numbers except 12 and -3.

Vertical Asymptotepronounced… “as-im-toht”

In a rational function R, if (x – a) is a factor of the denominator but not a factor of the numerator, x = a is vertical asymptote of the graph of R.

What is an asymptote?

•It is a line that a curve approaches but does not reach.

To find vertical asymptotes

1. Find the zeros of the denominator2. Factor numerator3. Simplify fraction4. There are vertical asymptotes at

any factors that are left in the denominator

Identify all vertical asymptotes of

2

3( )

3 2

xr x

x x

Step 1: Factor the denominator.

Step 2: Solve the denominator for x.

Equations for the vertical asymptotes are x = 2 and x = 1.

3( )

( 2)( 1)

xr x

x x

More Practice

2

2

6( )

9

x xf x

x

( 2)( 3)

( 3)( 3)

x x

x x

Identify the domain and any vertical

asymptotes. 2

2

6( )

9

x xf x

x

D: All Real #’s except x=-3,3

VA: at x=-3

2

3

x

x

Look at the table for this function:

2

2

6 ( 2)( 3) 2( )

9 ( 3)( 3) 3

x x x x xf x

x x x x

We can understand why the -3 shows an “error” message.

Buy why does the 3 also show an “error” message?

That means there is a “Hole” in the graph…

2

2

6 ( 2)( 3) 2( )

9 ( 3)( 3) 3

x x x x xf x

x x x x

That is what happens to the part we “cross

off” the fraction. That is where the hole(s) is.

Holes in GraphsIn a rational function R, if x – b is a factor of the numerator and the denominator, there is a hole in the graph of R when x = b (unless x = b is a vertical asymptote).

There is a vertical asymptote at x=-3.

And a hole at x=3.

Horizontal Asymptote

•If degree of P < degree of Q, thenthe horizontal asymptote of R is y = 0.

R(x) = is a rational function;

P and Q are polynomials

P

Q

2( )

2 3

x smallf x

x x bigger

So… HA: y=0

Horizontal Asymptote

R(x) = is a rational function;

P and Q are polynomials

P

Q

•If degree of P = degree of Q and a and b are the leading coefficients of P and Q, then

the horizontal asymptote of R is y = .

a

b2

2

16( )

4 5

x samef x

x x same

So… HA: y = 1

Horizontal Asymptote

Horizontal Asymptote

R(x) = is a rational function;

P and Q are polynomials

P

Q

•If degree of P > degree of Q, thenthere is no horizontal asymptote

3

2

7( )

4 3

x biggerf x

x x small

So… HA: D.N.E.

Horizontal Asymptotes

0

. . .

smallHA is y

bigger

same aHA is y

same bbigger

HA D N Esmall

Slant Asymptote

2 3 24 3 0 0 7x x x x x x

3 24 3x x x

A Slant Asymptote occurs when the degree of the numerator is exactly one degree higher than the degree of the denominator.3

2

7( )

4 3

x biggerf x

x x small

HA: D.N.E.

( ) 24 3x x

74

24 16 12x x ( ) 13 12x

Therefore:

Slant asymptote is

Y =

4x

Let . Identify the domain

and range of the function, all asymptotes and all

intercepts. Oh, also are there any holes?

3

2( )

20

xR x

x x

3

( )( 5)( 4)

xR x

x x

Equations for the vertical asymptotes are x = -5 and x = 4.

Because the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptotes, but slant asymptote is y = x - 1.

D: x ‡ -5, 4

R: ???

ONLY Intercept is ( 0 , 0 )

1Let . Find Domain & Range. Identify all asymptotes, holes and all Intercepts.

2

2

2 1( )

9

xR x

x

22 1( )

( 3)( 3)

xR x

x x

Vertical asymptotes: x = -3 and x = 3, but NO holesHorizontal asymptotes: 2

1leading coefficients

numerator and denominator have the same degree

y = 2

D: x ‡ 3, -3

R: y ‡

2

x -intercept: ( ½ √2, 0 ) ( -½ √2, 0 )

Y – intercept: ( 0 , 1/9 )

Identify all Critical Values in the graph of the rational function, then graph.

f(x) = 2x2 + 2x

x2 – 1factor: f(x) =

2x(x + 1)

(x + 1)(x – 1)

hole in the graph:x = –1

vertical asymptote:x = 1

horizontal asymptote:y = 2

D: x ‡ 1, -1

R: y ‡ 2

Intercepts:

( 0, 0 ) ( -1, 0)

To graph rational functions

1. Simplify function any restrictions should be listed.

2. Plot y intercept (if any)3. Plot x intercepts ( zeros of the top)4. Sketch all asymptotes (dash lines)5. Plot at least one point between each x intercept

and vertical asymptote6. Use smooth curves to complete graph

7.

For , identify all

Critical Values, then graph the function.

2 2 4( )

2 1

x xg x

x

D: Holes:

V.A.:

H.A.: R:

S.A.:

X-intercepts:

Y-intercepts:

For , identify all

Critical Values, then graph the function.

2

2

25( )

2 7 15

xg x

x x

D: Holes:

V.A.:

H.A.: R:

S.A.:

X-intercepts:

Y-intercepts:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

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