1. Graphing Rational Functions
Sect. 2.7
Graphing Rational Functions

2. Oblique asymptote at y = x + 5
ASYMPTOTES
If the degree of the numerator is greater than the degree of the denominator by 1, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder.
degree of top = 3
degree of bottom = 2
Oblique asymptote at y = x + 5

3. Find the oblique asymptote:
Equation of Asymptote!
Example 1:
Find the oblique asymptote:
Remainder is
Irrelevant

4. Degree of denominator = 2
Example 2:
Find the horizontal or
oblique asymptote:
Degree of numerator = 1
Degree of denominator = 2
Since the degree of the numerator is less than the degree of the denominator, there is NO oblique asymptote but a horizontal one at y = 0.

5. Degree of denominator = 1
Example 3:
Find the horizontal
or oblique asymptote:
Degree of numerator = 2
Degree of denominator = 1
Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
The oblique asymptote is found by long
division

6. Degree of denominator = 1
Example 4:
Find the horizontal
or oblique asymptote:
Degree of numerator = 1
Degree of denominator = 1
Since the degree of the numerator is equal to the degree of the denominator, there is NO oblique asymptote but a horizontal one at

7. Degree of denominator = 1
Example 5:
Find the horizontal
or oblique asymptote:
Degree of numerator = 3
Degree of denominator = 1
Since the degree of the numerator is differs from the degree of the denominator by more than 1, there is NO HA and NO OA.

8. Graphing a Rational Function
Suppose that where p(x) and q(x) are polynomial functions
with no common factors.
Calculate the x-intercepts by setting the function = 0 and
solving for x (This is the same as solving the equation p(x) = 0.).
2. Calculate the y-intercept by setting x = 0 and solving for y.
2. Find vertical asymptote(s) by factoring the numerator/ denominator, cancelling like factors, and solving the equation q(x) = 0.
3. Find holes by taking the factors that cancelled, setting them = 0, and solving for x.
4. Find the horizontal/oblique asymptote(s) using the rule based on the degrees of p(x) and q(x).
5. Plot points in between the x-intercepts and sketch the graph labeling the asymptotes.

9. Example 1: Sketch the graph of
The vertical asymptote is x = -2.
The horizontal asymptote is y = 2/5.
The x-intercept is x = 3/2.
The y-intercept is -3/10.

10. Graphing a rational function where m = n
3x2
x2-4
Ex 2: Graph y =
x-intercepts: Set top = 0 and solve!
3x2 = 0  x2 = 0  x = 0.
Vertical asymptotes: Set bottom = 0 and solve
x2 - 4 = 0  (x - 2)(x+2) = 0  x= ±2
Degree of p(x) = degree of q(x)  top and bottom tie  divide the leading coefficients
Y = 3/1 = 3.
Horizontal Asymptote: y = 3

11. You’ll notice the three branches.
Here’s the picture! x y -4 4 -3
5.4 -1 1 3
You’ll notice the three branches.
This often happens with overlapping horizontal and vertical asymptotes.
The key is to test points in each region!
Domain:
x ≠ ±2
Range:
y > 3 & y ≤ 0

12. Graphing a Rational Function where m > n
Ex 3: Graph y =
x-intercepts: x2- 2x - 3 = 0
(x - 3)(x + 1) = 0  x = 3, x = -1
Vertical asymptotes: x + 4 = 0  x = -4
Degree of p(x) > degree of q(x)
No horizontal asymptote
OA – use synthetic division y = x - 6
x2- 2x - 3
x + 4

13. Picture time! x y -12 -20.6 -9 -19.2 -6 -22.5 -2 2.5 -0.75 2 -0.5 6
-0.75 2
-0.5 6
2.1

14. Ex 4 Graph y =
domain: all real numbers
x-intercepts: None; p(x) = 6 and 6 ≠ 0
Vertical Asymptotes: None;
x2 + 3 = 0  x2 = -3. No real solutions.
Degree p(x) < Degree q(x) --> Horizontal Asymptote at y = 0 (x-axis) 6
x2 + 3

15. x y 2 -1
1.5 1 -2
6/7
y = 6
x2 + 3

16. Horizontal Asymptotes:
Ex 5
Domain:
Vertical Asymptotes:
Horizontal Asymptotes:
Holes:
Intercepts:

17. Horizontal Asymptotes:
Ex 6
Domain:
Vertical Asymptotes:
Horizontal Asymptotes:
Holes:
Intercepts:

18. Graph Ex 7 x y -2 -.4 -1 -.5 0 0 1 .5 2 .4 y-intercepts: 0
x-intercepts: x=0
vert. asymp.: x2+1=0
x2= -1
No vert asymp
4. horiz. asymp: 1<2
(deg. of top < deg. of bottom) y=0
x y
(No real solns.)

19. Domain: (-∞,∞)
Range:

20. Ex 8
Graph
slant asymptote: y = –2x – 4

21. Horizontal Asymptotes:
Ex 9
Domain:
Vertical Asymptotes:
Horizontal Asymptotes:
Holes:
Intercepts:

22. The Big Ideas Always be able to find:
x-intercepts (where numerator = 0)
Vertical asymptotes (where denominator = 0)
Horizontal/Oblique asymptotes:
If bottom wins: x-axis asymptote
If they tie: divide leading coeff.
If top wins: No horizontal asymp
Oblique asym (use long division).
Sketch branch in each region (plot points)

23. Closure
Explain the process for calculating the horizontal and oblique asymptotes of a rational function.