1. 8.1/8.2- Graphing Rational Functions
WILL TAKE THE FULL CLASS!!!

2. Definition Parent reciprocal function: 𝑓 𝑥 = 1 𝑥 , where 𝑥≠0 x y 1 210
1/2 -1 -2
-10
-1/2

3. Definition (cont.)
Vertical asymptote: Imaginary vertical line which you can NEVER cross
Horizontal asymptote: Imaginary horizontal line which you can POSSIBLY cross
Parent reciprocal function
VA: 𝑥=0
HA: 𝑦=0

4. Transformations VA: 𝒙=𝒉 HA: 𝒚=𝒌 𝑦= 𝑎 𝑥−ℎ +𝑘
𝑎: Reflection or vertical stretch or shrink
𝑎<0: reflection over x-axis
𝑎>1: vertical stretch
0<𝑎<1: vertical compression
ℎ: Horizontal translation
Minus in the middle means go right
Plus in the middle means go left
𝑘: Vertical translation
Plus means go up
Minus means go down
VA: 𝒙=𝒉
HA: 𝒚=𝒌

5. Ex: For 𝒚= 𝟖 𝒙 , find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int:
Y-int:
Domain:
Range:

6. On Your Own: For 𝒚= 𝟑 𝒙 , find:
𝑎= ℎ= 𝑘=
Transformation:
VA:
HA:
Graph.
X-int:
Y-int:
Domain:
Range:

7. Ex: For 𝒚= −𝟐 𝒙 , find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int:
Y-int:
Domain:
Range:

8. Ex: For 𝒚= 𝟏 𝒙+𝟏 −𝟐, find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph.
X-int:
Y-int:
Domain:
Range:

9. On Your Own: For 𝒚= 𝟏 𝒙−𝟒 +𝟔, find:
𝑎= ℎ= 𝑘=
Transformation:
VA:
HA:
Graph.
X-int:
Y-int:
Domain:
Range:

10. Answer: (A) because it moved 3 left (h = -3) and 4 up (k = 4)
Ex:
Answer: (A) because it moved 3 left (h = -3) and 4 up (k = 4)

11. On Your Own:
This graph of a function is a translation of the graph of 𝑦= 2 𝑥 . What is an equation for the function?
Answer: 𝑦= 2 𝑥−1 −4 because it moved 1 right (h = 1) and 4 down (k = -4)

12. Def Rational functions: Fractions of polynomial functions
When is a fraction undefined?
Domain of a rational function = all values except when you cancel a factor or when the denominator is 0.
If you have a denominator left over, the value that makes that denominator 0 makes a vertical asymptote

13. Continuous vs. Discontinuous
Continuous: When you cannot cancel a factor and the denominator is never 0
This means when you draw the graph, your pencil never leaves the paper
Ex: 1st graph
Discontinuous: When you cancel a factor or the denominator can be 0
This means when you draw the graph, your pencil must leave the paper
Ex: 2nd and 3rd graph

14. Discontinuities If you cancel a factor, you create a hole at that spot
Ex: The graph of 𝑦= (𝑥+3)(𝑥+2) 𝑥+2 has a hole at 𝑥=−2
Holes are also called removable discontinuities
If you still have a denominator left over, you have a vertical asymptote at that spot
Ex: The graph of 𝑦= 𝑥+4 𝑥−2 has a vertical asymptote at 𝑥=2
START FROM HERE NEXT TIME

15. Domain= all real numbers except 𝑥=3 and 𝑥=1 (VA’s)
Find the domain of each rational function. Identify what occurs at those points. Find the x & y-intercepts.
a) 𝑦= 𝑥+3 𝑥 2 −4𝑥+3
Answers:
Domain= all real numbers except 𝑥=3 and 𝑥=1 (VA’s)
x-int = (-3 , 0)
y-int = (0 , 1)

16. Domain= all real numbers except 𝑥=4 (hole)
Find the domain of each rational function. Identify what occurs at those points. Find the x & y-intercepts.
b) 𝑦= 𝑥 2 −3𝑥−4 𝑥−4
Answers:
Domain= all real numbers except 𝑥=4 (hole)
x-int = (-1 , 0)
y-int = (0 , 1)

17. Domain= all real numbers
Ex:
Find the domain of each rational function. Identify what occurs at those points. Find the x & y-intercepts.
c) 𝑦= 𝑥−5 𝑥 2 +1
Answers:
Domain= all real numbers
x-int = (5 , 0)
y-int = (0 , -5)

18. Horizontal Asymptotes
To find a HA, compare the degree of the numerator and denominator
If the degree of the numerator is m and the degree of the denominator is n, then either of these can happen:
For 𝑚<𝑛, HA at 𝑦=0 (the x-axis)
For 𝑚=𝑛, HA at 𝑦= 𝑎 𝑏 where a is the coefficient of the leading term in the numerator and b is the coefficient of the leading term in the denominator
For 𝑚>𝑛, no HA

19. Ex: What is the horizontal asymptote for:
𝑦= 2𝑥 𝑥− b) 𝑦= 𝑥−2 𝑥 2 −2𝑥−3 c) 𝑦= 𝑥 2 2𝑥−5
Answers:
𝑦=2
𝑦=0
No HA

20. Ex: For 𝑦= 𝑥+4 𝑥−4 , Hole = ________ VA: __________ HA: __________
X-int = _______
Y-int = _______
Graph the rational function

21. Ex: For 𝑦= 𝑥+6 𝑥 2 +𝑥−6 , Hole = ________ VA: __________
HA: __________
X-int = _______
Y-int = _______
Graph the rational function

22. Ex: For 𝑦= 𝑥 2 −4 3𝑥−6 , Hole = ________ VA: __________ HA: __________
X-int = _______
Y-int = _______
Graph the rational function

23. Ex:

24. Ex: