graphing rational functions. copyright © by houghton mifflin company, inc. all rights reserved....

Graphing Rational Functions

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

x f(x)

2 0.5

1 1

0.5 2

0.1 10

0.01 100

0.001 1000

x f(x)

-2 -0.5

-1 -1

-0.5 -2

-0.1 -10

-0.01 -100

-0.001 -1000

As x → 0–, f(x) → -∞.As x → 0+, f(x) → +∞.

A rational function is a function of the form f(x) = ,

where P(x) and Q(x) are polynomials and Q(x) = 0.)(

)(

xQ

xP

f(x) =x

1

Example: f (x) = is defined for all real numbers except x = 0.x

1


x

x = a

as x → a –

f(x) → + ∞

x

x = a

as x → a –

f(x) → – ∞

x

x = a

as x → a +

f(x) → + ∞

x

x = a

as x → a +

f(x) → – ∞

The line x = a is a vertical asymptote of the graph of y = f(x),


Example: Show that the line x = 2 is a vertical asymptote of the

graph of f(x) = .2)2(

4

x

x f(x)

1.5 16

1.9 400

1.99 40000

2 -

2.01 40000

2.1 400

2.5 16

Observe that:x→2–, f (x) → – ∞

x→2+, f (x) → + ∞

This shows that x = 2 is a vertical asymptote.

y

x100

0.5

f (x) = 2)2(

4

x

x = 2


Set the denominator equal to zero and solve.

Solve the quadratic equation x2 + 4x – 5.

(x – 1)(x + 5) = 0Therefore, x = 1 and x = -5 are the values of x

for which f may have a vertical asymptote.

Example:Find the vertical asymptotes of the graph of f(x) = .

)54(

12 xx


1. Find the roots of the denominator. 0 = x2 – 4 = (x + 2)(x – 2) Possible vertical asymptotes are x = -2 and x = +2.2. Since (x+2) occurs on both the top and on bottom of the function, it will cancel, which will leave a hole in the graph. You must plug -2 back into the function to calculate the y value.

f is undefined at -2

A hole in the graph of f at (-2, -0.25) shows a removable singularity.

x = 2

Example: Find the vertical asymptotes of the graph of f(x) = .

)4(

)2(2

x

x

x

y

(-2, -0.25)


y

y = b

as x → + ∞ f(x) → b –

y

y = b

as x → – ∞ f(x) → b –

y

y = b

as x → + ∞ f(x) → b +

y

y = b

as x → – ∞ f(x) → b +

The line y = b is a horizontal asymptote of the graph of y = f(x)


x f(x)

10 0.1

100 0.01

1000 0.001

0 –-10 -0.1

-100 -0.01

-1000 -0.001

As x becomes unbounded positively, f(x) approaches zero from above; therefore, the line y = 0 is a horizontal asymptote of the graph of f. As f(x) → – ∞, x → 0 –.

Example: Show that the line y = 0 is a horizontal asymptote of the graph of the function f(x) = .

x

1

x

y

f(x) =x

1

y = 0


y

x

Similarly, as x → – ∞, f(x) →1–.

Therefore, the graph of f has y = 1 as a horizontal asymptote.

Example: Determine the horizontal asymptotes of the graph of

f(x) = .)1( 2

2

x

x

As x → +∞, → 1–.)1( 2

2

x

x

y = 1


Finding Asymptotes for Rational Functions

• If c is a real number which is a root of both P(x) and Q(x), then there is a removable singularity at c.

• If c is a root of Q(x) but not a root of P(x), then x = c is a vertical asymptote.

• If m > n, then there are no horizontal asymptotes.

• If m < n, then y = 0 is a horizontal asymptote.

• If m = n, then y = am is a horizontal asymptote.bn

Given a rational function: f (x) = P(x) am xm + lower degree terms

Q(x) bn xn + lower degree terms=


Factor the numerator and denominator.

The only root of the numerator is x = -1. The roots of the denominator are x = -1 and x = 2 .

Since -1 is a common root of both, there is a hole in the graph at -1 .

Since 2 is a root of the denominator but not the numerator, x = 2 will be a vertical asymptote.

Since the polynomials have the same degree, y = 3 will be a horizontal asymptote.

Example: Find all horizontal and vertical asymptotes of f (x) = .

2

3632

2

xx

xx

y = 3

x = 2

x

y


Example

2

7( )

3

xf x

x

For each function, state the following:a.The domainb.Vertical asymptotesc.Horizontal asymptotes


Example

2

2

6 4 7( )

5 2

x xf x

x

For each function, state the following:a.The domainb.Vertical asymptotesc.Horizontal asymptotes


Practice Time

Do #’s 1-6 on the worksheet


Example

3( )

2f x

x

Sketch the graph of each function. Includea.All interceptsb.Vertical and horizontal asymptotesc.Any other necessary points


Example

2( )

3

xf x

x



Example

2

5( )

6

xf x

x x



Practice Time

Do #’s 7 - 11 on the worksheet


A slant asymptote is an asymptote which is not vertical or horizontal.

The slant asymptote is y = 2x – 5.

As x → + ∞, → 0+. 3

14

x

Example: Find the slant asymptote for f(x) = .3

12 2

x

xx

x

yx = -3

y = 2x - 5

Divide:3

12 2

x

xx3

1452

xx

Therefore as x → ∞, f(x) is more like the line y = 2x – 5.

3

14

xAs x → – ∞, → 0–.


Example

22 1( )

xf x

x

Sketch the graph of each function. Includea.All interceptsb.Vertical and slant asymptotesc.Any other necessary points


Example

2

( )1

xf x

x

Sketch the graph of each function. Includea.All interceptsb.Vertical and slant asymptotesc.Any other necessary points


Practice Time

Do #’s 12 - 16 on the worksheet

graphing rational functions. copyright © by houghton mifflin company, inc. all rights reserved....

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