Sec 4.3

Graphing Rational Functions

Math 1051 - Precalculus I

Graphing Rational Functions Sec 4.3

Sec 4.2 Graphing Rational Functions

Find the asymptotes of

f (x) =x3 + 2x2 + x

x2 − 3x

Ans: x = 3 and y = x + 5

Graphing Rational Functions Sec 4.3

Sec 4.2 Graphing Rational Functions

Find the asymptotes of

f (x) =x3 + 2x2 + x

x2 − 3x

Ans: x = 3 and y = x + 5

Graphing Rational Functions Sec 4.3

How to graph rational functions

We’ve done most of the prep work, now we just have to put it alltogether.

Let’s do it with an example...

Graphing Rational Functions Sec 4.3

How to graph rational functions

We’ve done most of the prep work, now we just have to put it alltogether.

Let’s do it with an example...

Graphing Rational Functions Sec 4.3

How to graph rational functions

We’ve done most of the prep work, now we just have to put it alltogether.

Let’s do it with an example...

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):

1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.

2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?

3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?

4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”

5 Plot places where your graph crosses the horizontal oroblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)

7 Connect the dots

Graphing Rational Functions Sec 4.3

f (x) =x + 2x2 − 9

Steps (very similar to graphing polynomials):1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

R(x) =x + 2x2 − 9

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Graphing Rational Functions Sec 4.3

R(x) =x2 + 5x + 6

x + 1

1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

R(x) =x2 + 5x + 6

x + 1

-6 -4 -2 2 4 6

-5

5

10

Graphing Rational Functions Sec 4.3

R(x) =x2 + 2x + 1

x2 − 1

1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

R(x) =x2 + 2x + 1

x2 − 1

-6 -4 -2 2 4 6

-5

5

10

Graphing Rational Functions Sec 4.3

R(x) =x2 + 2x + 1

x2 − 1

-6 -4 -2 2 4 6

-5

5

10

Don’t forget the hole at (−1,0)Graphing Rational Functions Sec 4.3

R(x) =x3

x2 − 2x + 1

1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

R(x) =x3

x2 − 2x + 1

-10 -5 5 10

-10

-5

5

10

Graphing Rational Functions Sec 4.3

R(x) =2(x − 2)(x + 3)(x − 4)

x(x − 9)2

1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

R(x) =2(x − 2)(x + 3)(x − 4)

x(x − 9)2

-10 -5 5 10 15 20 25

-10

10

20

30

Graphing Rational Functions Sec 4.3

R(x) =−1

x − 2+ 3

1 Factor, state domain, THEN reduce.2 Plot intercepts. Touch or cross?3 Vertical asymptotes. “Touch or cross”?4 Horizontal/oblique asymptote = “End behavior”5 Plot places where your graph crosses the horizontal or

oblique asymptote, if it does. Solve:

Function = AsymptoteR(x) = y

6 Plot a few extra points (one in each “interval”)7 Connect the dots

Graphing Rational Functions Sec 4.3

R(x) =−1

x − 2+ 3

-10 -5 5 10

-10

-5

5

10

Graphing Rational Functions Sec 4.3

R(x) =−1

x − 2+ 3

Is there another way we can graph this function?

Graphing Rational Functions Sec 4.3

Going backwards. Find a function that has this graph:

-10 10 20

-10

-5

5

10

Graphing Rational Functions Sec 4.3

Read section 4.4 for Monday.

Graphing Rational Functions Sec 4.3