Assignment 1b- N

Nermin Fialkowski Dr. Monica Kelly AIL 622 8 September, 2018 Assignment 1B- Set Up Essential Questions 1. What other properties of functions can I use to help provide me with a better understanding of how these functions behave? 2. How can limits provide me with insight to convergent and divergent functions? Chapter 5: Section 1- Rational Functions Word Wall Rational Function

Reciprocal Function

Vertical Asymptotes

Slant Asymptotes

Horizontal Asymptotes

Approach Rational Functions !

Rewriting rational functions into transformations of ". Use idea of rewriting improper fractions into mix numbers to achieve this. 20 17 + 3 3 = =1+ 17 17 17

𝑥+2 𝑥−1+2+1 𝑥−1+3 𝑥−1 3 = = = + 𝑥−1 𝑥−1 𝑥−1 𝑥−1 𝑥−1 =1+

3 3 → +1 𝑥−1 𝑥−1

0(")

Vertical asymptotes of rational functions 𝑓(𝑥) = 1(") 𝑞(𝑥) = 0

3" 4 5⋯

Horizontal asymptotes of rational functions 𝑓(𝑥) = 7" 8 5⋯ If 𝑛 < 𝑚

If 𝑛 = 𝑚

If 𝑛 > 𝑚

Horizontal Asymptote = 0

Horizontal Asymptote = 7

3

No Horizontal Asymptote

If 𝑛 is greater than 𝑚 by EXACTLY 1, then there is a Slant Asymptote → Do long division

Reciprocal Functions !

Essential Question #1

Graph 𝑓(𝑥) = 𝑥 ? − 4 and 𝑔(𝑥) = B(") on the same axis Use properties of 𝑓(𝑥), to graph 𝑔(𝑥) Why/how do the zeros of 𝑓(𝑥) turn into the vertical asymptotes of 𝑔(𝑥)? What remains positive/negative? Increasing/decreasing?

Differentiated Learning 0(")

Have students create their own rational 𝑓(𝑥) = 1(")

Will use all of their knowledge on zeros, asymptotes, y-intercept, and re-writing !

into transformations of ", in order to create their own unique graphs. Will be put on display Chapter 5: Section 2- Limits Word Wall Limit

Continuous

Hole

Vertical

Slant Asymptote

Asymptote Horizontal

Infinity

Does Not Exist

One-Sided Limit

Asymptote

Piecewise Functions

Approach Limits Intuitive approach using Benny & Bertha the Bug

Essential Question #2

As Benny & Bertha “get closer, and closer” to said x-value, how high are they getting (yvalue)? Will Benny & Bertha meet at the same place? Use for both One-Sided Limit and Definition of a Limit The existence of a point is irrelevant for a limit to be possible. What matters is where Benny & Bertha go and if they go to the same place 𝑓(𝑎) does not have to equal lim 𝑓(𝑥) "→3

Algebracially solving a limit Direct Substition Numberical Value ! H H H

→Vertical Asysmptote →Indeterminate→Do “algebra” Factor Hole Rationalize

Formal definition of continutity 𝑓(𝑎) exists lim 𝑓(𝑥) exists

"→3

lim 𝑓(𝑥) = 𝑓(𝑎)

"→3

Piecewise Function Use to discuss continuty, evalute limits, and identify y-values.

Differentiated Learning Students will create their own piecewise graphs (like above) and pose various questions about the graph. Will quiz their classmates on the answers. 𝑓(−1)

𝑓(1)

lim 𝑓(𝑥)

"→!

lim 𝑓(𝑥)

"→I

lim 𝑓(𝑥)

"→JI

lim 𝑓(𝑥)

"→JK

𝑓(−4)