1. Grab a calculator and graph the following equations:
Starter – Day December 11
Content Objective: SWBAT graph and shift rational functions.
Language Objective: SWBAT explain how to determine the vertical and horizontal shifts of a rational function.
Grab a calculator and graph the following equations:
1. y= 1 𝑥 y= 1 𝑥 𝑦= 1 𝑥 +2

2. Each input has a single output —like a vending machine.
Function
Each input has a single output —like a vending machine.
Often written as "𝑓 𝒙 " where 𝒙 is the input value.
Example: 𝑓 𝒙 =𝟐𝒙
𝑊ℎ𝑒𝑛 𝒙=𝟑, 𝑓 𝟑 =𝟐 𝟑
𝑓 𝟑 =𝟔

3. The domain of 𝑓 𝒙 is EXCEPT values that make the denominator 0.
Rational Function
𝑓 𝒙 = 𝒙 𝒙+2
A rational expression (a fraction with a polynomial in the numerator and the denominator) that has an equal sign and 𝑓 𝒙 on the other side.
The domain of 𝑓 𝒙 is EXCEPT values that make the denominator 0.

4. Reciprocal Function
𝑓 𝒙 = 1 𝒙

5. Asymptote
A line that a graph approaches (but never reaches) as it heads toward infinity.
graph

6. Types of Asymptotes
Horizontal Asymptotes
Vertical Asymptotes

7. 𝒂 indicates a stretch or shrink of the graph.
Rules for Graphing Rational Functions
𝒚= 𝒂 𝒙−𝒉 +𝒌
𝒂 indicates a stretch or shrink of the graph.
If 𝒂 is negative, the graph reflects over the x-axis.
𝒉 indicates a horizontal shift (left or right).
𝒌 indicates a vertical shift (up or down).
The line 𝒙=−𝒉 is a vertical asymptote.
The line 𝒚=𝒌 is a horizontal asymptote.

8. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Identify each piece of this equation, and then graph:
𝑦= 1 𝑥−2 +3
𝒙=−𝒉
𝒚=𝒌

9. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Identify each piece of this equation, and then graph:
𝑦= 1 𝑥−1 +4

10. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Identify each piece of this equation, and then graph:
𝑦= 1 𝑥−0 −1

11. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Identify each piece of this equation, and then graph:
𝑦= 2 𝑥 −3

12. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Write a function that shifts 𝑦= 1 𝑥 up 2 and left 4.

13. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Write a function that shifts 𝑦= 1 𝑥 down 3 and right 1.

14. 𝒚= 𝒂 𝒙−𝒉 +𝒌
Write an equation for the following graph:

15. LAB Writing
Using explain how to determine the vertical and horizontal shifts of a rational function.
Assignment
Book Section 3.5 No. 1-16, 25, 29, 35
𝒚= 𝒂 𝒙−𝒉 +𝒌