Sec 4hankx003/Fall2012/Lectures/Ch4Sec3.pdf · 4 Horizontal/oblique asymptote = “End behavior” 5 Plot places where your graph crosses the horizontal or oblique asymptote, if it

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


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...


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


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





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





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





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:




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





f (x) =x + 2x2 − 9




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

7 Connect the dots


f (x) =x + 2x2 − 9






R(x) =x + 2x2 − 9

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6


R(x) =x2 + 5x + 6

x + 1






R(x) =x2 + 5x + 6

x + 1

-6 -4 -2 2 4 6

-5

5

10


R(x) =x2 + 2x + 1

x2 − 1






R(x) =x2 + 2x + 1

x2 − 1

-6 -4 -2 2 4 6

-5

5

10


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






R(x) =x3

x2 − 2x + 1

-10 -5 5 10

-10

-5

5

10


R(x) =2(x − 2)(x + 3)(x − 4)

x(x − 9)2






R(x) =2(x − 2)(x + 3)(x − 4)

x(x − 9)2

-10 -5 5 10 15 20 25

-10

10

20

30


R(x) =−1

x − 2+ 3






R(x) =−1

x − 2+ 3

-10 -5 5 10

-10

-5

5

10


R(x) =−1

x − 2+ 3

Is there another way we can graph this function?


Going backwards. Find a function that has this graph:

-10 10 20

-10

-5

5

10


Read section 4.4 for Monday.


Sec 4hankx003/Fall2012/Lectures/Ch4Sec3.pdf · 4 Horizontal/oblique asymptote = “End behavior” 5 Plot places where your graph crosses the horizontal or oblique asymptote, if it

Documents