1. Warm-up
Solve the following rational equation.

2. Set Equation to ZERO
Next Slide

3. Problem Continued
MUST CHECK ANSWERS
x = -4 does not work

4. Properties of Rational Function and Discontinuities

5. Objectives I can identify the domain and range
I can identify Graph Discontinuities
Vertical Asymptotes
Horizontal Asymptotes
Slant/Oblique Asymptotes
Holes
I can find “x” and “y” intercepts
I can find Interval of Increase and Decrease

6. Properties of Rational Functions
Defn:
Rational Function
The functions p and q are polynomials.
The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero. The range is the set of all real numbers except the values of the horizontal asymptote and hole.

7. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
{x | x  –4} or
(-, -4)  (-4, )

8. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
{x | x  2} or
(-, 2)  (2, )

9. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
{x | x  –3, 3} or
(-, -3)  (-3, 3)  (3, )

10. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
{x | x  –3, 5} or
(-, -3)  (-3, 5)  (5, )

11. Asymptotes
Asymptotes are the boundary lines that a rational function approaches, but never crosses.
We draw these as Dashed Lines on our graphs.
There are three types of asymptotes:
Vertical
Horizontal (Graph can cross these)
Slant

12. Vertical Asymptotes
Vertical Asymptotes exist where the denominator would be zero.
They are graphed as Vertical Dashed Lines
There can be more than one!
To find them, set the denominator equal to zero and solve for “x”

13. Example #1
Find the vertical asymptotes and domain for the following function:
Set the denominator equal to zero
x – 1 = 0, so x = 1
This graph has a vertical asymptote at x = 1
Domain: {x | x ≠ }

14. y-axis Vertical Asymptote at X = 1 x-axis 9 8 7 6 5 4 3 2 10 -9 -8 -7-6 -5 -4 -3 -2 -1 1
x-axis -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9

15. Other Examples:
Find the vertical asymptotes and domain for the following functions:

16. To find Vertical Asymptote(s)
1) Set reduced denominator = 0
Solve for x = #.
Your answer is written as a line.

17. Horizontal Asymptotes
Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary.
To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator
See next slide:

18. Horizontal Asymptote (HA)
Given Rational Function:
Compare DEGREE of Numerator to Denominator
If N < D , then y = 0 is the HA
If N > D, then the graph has NO HA (maybe a slant)
If N = D, then the HA is

19. Example #1
Find the horizontal asymptote and range for the following function:
Since the degree of numerator is equal to degree of denominator (m = n)
Then HA: y = 1/1 = 1
This graph has a horizontal asymptote at y = 1
Range {y | y ≠ 1}

20. Horizontal Asymptote at y = 1
y-axis 9 8 7
Horizontal Asymptote at
y = 1 6 5 4 3 2 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1
x-axis -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9

21. Other Examples:
Find the horizontal asymptote and range for the following functions:

22. To find Horizontal Asymptote(s)
1) Compare DEGREE of numerator and denominator
Num BIGGER then NO HA
Num SMALLER then y = 0
Degree is SAME then

23. Slant Asymptotes (SA)
Slant asymptotes exist when the degree of the numerator is one larger than the denominator.
Cannot have both a HA and SA
To find the SA, divide the Numerator by the Denominator.
The results is a line y = mx + b that is the SA.

24. Example of SA -2

25. To find Slant Asymptote(s)
1) DEGREE of Numerator must be ONE bigger than Denominator
Divide with Synthetic or Long Division
Don’t use the Remainder
Get y = mx + b

26. Holes
A hole exists when the same factor exists in both the numerator and denominator of the rational expression and the factor is eliminated when you reduce!

27. Example of Hole Discontinuity

28. HOLES To Find Holes 1) Factor. 2) Reduce.
A hole is formed when a factor is eliminated from the denominator.
Set eliminated factor = 0 and solve for x.
5) Find the y-value of the hole by substituting the x-value into reduced form and solve for y.
6) Your answer is written as a point. (x, y)

29. To find x- intercept(s)
x-intercepts  when y = 0
Set reduced numerator = 0
2) Solve for x.
3) Answer is written as a point.
(#, 0)

30. To find y- intercept
y – intercepts  when x = 0
1) Substitute 0 in for all x’s in reduced form.
Solve for y.
Answer is a point. (0, #)

31. End Behavior
As the function approaches - ∞ and + ∞, what y-value is the function approaching?
You want to look at the FAR left and right of the graph.
If the graph is going up, the y is approaching + ∞.
If the graph is going down, then y is approaching - ∞.
If there is a HORIZONTAL ASYMPTOTE,
then the y – values should be “flattening out” to the asymptote’s value.

32. Intervals of Increasing and Decreasing
These are intervals on the x-axis where the function rises (increase) and falls (decrease). Imagine tracing the curve of the function from LEFT TO RIGHT. If it traces in an upward direction/climbing, the function is increasing. If it traces in a downward direction/falling, the function is decreasing.
Hint: The vertical asymptotes help break up the intervals.

33. Intervals of Increasing and Decreasing
Increase
 
Decrease
Increase
 

34. Practice: For each rational function, identify the range, intervals of increase/decrease AND end behavior. 1)
Domain:
VA:
Range:
HA:
SA:
Intervals of Increase:
Intervals of Decrease:
End Behavior:

35. Practice: For each rational function, identify the range, intervals of increase/decrease AND end behavior. 2)
Domain:
VA:
Range:
HA:
SA:
Intervals of Increase:
Intervals of Decrease:
End Behavior:

36. Practice: For each rational function, identify the range, intervals of increase/decrease AND end behavior. 3)
Domain:
VA:
Range:
HA:
SA:
Intervals of Increase:
Intervals of Decrease:
End Behavior:

37. Video Link(s) https://www.youtube.com/watch?v=yKTiaUT0nTI

38. Video Link(s) continues