RATIONAL

FUNCTIONS II

GRAPHING RATIONAL FUNCTIONS

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Gra

phin

g

Ratio

nal F

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ions

Find the domain. Excluded values are where your vertical asymptotes are.

Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x

Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving.

Test for symmetry by putting –x in for x. (remember even, odd test)

Find horizontal or oblique asymptote by comparing degrees

Find some points on either side of each vertical asymptote

Connect points and head towards asymptotes.

Find the domain. Excluded values are where your vertical asymptotes are.

6

62

xx

xR

062 xx

023 xx

2,3 so xx

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x

1600

60

2

R

So let’s plot the y intercept which is (0, - 1)

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

If the numerator of a fraction = 0 then the whole fraction = 0 since 0 over anything = 0

Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving.

But 0 = 6 is not true which means there IS NO x intercept.

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Not the original and not negative of function so neither even nor odd.

Test for symmetry by putting –x in for x. (remember even, odd test)

6

62

xx

xR

6

62

xx

xR

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find horizontal or oblique asymptote by comparing degrees

degree of the top = 0

0xremember

x0 = 1

degree of the bottom = 2

If the degree of the top is less than the degree of the bottom the x axis is a horizontal asymptote.

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find some points on either side of each vertical asymptote

x R(x)

Choose an x on the left side of the vertical asymptote.

-4

4.014

6

644

64 2

R

0.41

16

6

611

61 2

R

-1

Choose an x in between the vertical asymptotes.

Choose an x on the right side of the vertical asymptote.

4

16

6

644

64 2

R

1

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Connect points and head towards asymptotes.

There should be a piece of the graph on each side of the vertical asymptotes.

Pass through the point and head towards asymptotes

Pass through the points and head towards asymptotes. Can’t go up or it would cross the x axis and there are no x intercepts there.

Pass through the point and head towards asymptotes

Go to a function grapher or your graphing calculator and see how we did.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

The window on the calculator was set from -8 to 8 on both x and y.

Notice the calculator draws in part of the asymptotes, but these ARE NOT part of the graph. Remember they are sketching aids---the lines that the graph heads towards.

Find the domain. Excluded values are where your vertical asymptotes are.

9

342

2

x

xxxR

092 x

033 xx

3,3 so xx

Let's try another with a bit of a "twist":

But notice that the top of the fraction will factor and the fraction can then be reduced.

33

13

xx

xx

We will not then have a vertical asymptote at x = -3, but it is still an excluded value NOT in the domain.

vertical asymptote from this factor only since other factor cancelled.

3

1

x

xxS

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x

3

1

30

100

S

So let’s plot the y intercept which is (0, - 1/3)

We'll graph the reduced fraction but we must keep in mind that x - 3

3

1

x

xxS

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

If the numerator of a fraction = 0 then the whole fraction = 0 since 0 over anything = 0

Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving.

x + 1 = 0 when x = -1 so there is an x intercept at the point (-1, 0)

3

1

x

xxS

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Not the original and not negative of function so neither even nor odd.

Test for symmetry by putting –x in for x. (remember even, odd test)

3

1

x

xxS

3

1

x

xxS

3

1

x

xxS

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find horizontal or oblique asymptote by comparing degrees

degree of the top = 1

degree of the bottom = 1

If the degree of the top equals the degree of the bottom then there is a horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom.

11

1y

1

1

3

1

x

xxS

Find some points on either side of each vertical asymptote

x S(x)

4 5

Let's choose a couple of x's on the right side of the vertical asymptote.

51

5

34

144

S

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

We already have some points on the left side of the vertical asymptote so we can see where the function goes there

3.23

7

36

166

S

6 2.3

3

1

x

xxS

Connect points and head towards asymptotes.

There should be a piece of the graph on each side of the vertical asymptote.

Pass through the points and head towards asymptotes

Pass through the point and head towards asymptotes

Go to a function grapher or your graphing calculator and see how we did.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

REMEMBER that x -3 so find the point on the graph where x is -3 and make a "hole" there since it is an excluded value.

3

1

33

133

S

The window on the calculator was set from -8 to 8 on both x and y.

Notice the calculator drew the vertical asymptote but it did NOT show the "hole" in the graph. It did not draw the horizontal asymptote but you can see where it would be at y = 1.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8