The big thaw...

After our cold week off, I want you to assemble and bring in the text homework you have completed so far to show what you know about exponentials.  Next class you will be handed a highlighter and need to choose and highlight about 10 problems that show you know the learning targets listed below.  An even better plan is to highlight (I mean just draw a yellow box around significant problems to draw attention to important problems) before you come to the next class.

The problems you highlight should demonstrate that you know the learning targets described below:

Learning Targets            By the end of this unit you should be able to say and demonstrate that:..

(1)  I understand all the parameters of an exponential function y = Ca^x .   Given a table, I can determine initial values and growth/decay factors to write an exponential function.  Given a graph or situation, I can identify and interpret the appropriate equation using my understanding of exponentials.

(2)  Given a real-life scenario, I can create and evaluate exponential (and later logarithmic) functions to solve problems.  I can interpret different types of compounded interest formulas and y = e^x and use technology to identify solutions to problems.

 

(3)  I recognize a logarithm function as the inverse of an exponential function.  I can solve exponential equations by applying log base 10 and the natural log functions to “undo” exponential functions. 

 There will also be a survey you need to complete next class that rates your understanding of the targets above more specifically.  Be sure you have your work to look at so you can back up what you see about your understanding.

Second week of Exponentials and Missy Elliot

Last Week
We learned exponential growth and decay function key features and  end behavior.

- The y-intercept (0,a) is always a good feature to inspect and 
the end behavior either grows rapidly towards infinity or approaches the x-axis asymptotically.  [Unless the function is transformed of course]

- We learned the basic structure of exponential equations:  y=ab^x and y = a (1+r)^x
and it is all about the base (if "it" means determining growth versus this decay.

Next Step(s):  Apply and interpret more sophisticated exponential models, like models for compounded interest and models to display scientific phenomenon.

- We learned how to interpret patterns (mainly from tables) and determine if it represents an exponential or linear function.  Then, actually write the formula using the appropriate structure of an equation.

Next Step(s):  Contrast exponential functions with Quadratic Functions, and learn about the Exponential Function's Evil Twin Function which can undo it.  If we can undo an operation, we can solve equations.  

Mathematician's call it the Inverse Function, and Missy Elliot refers to it in one of her songs, Work it.  If it is worth it, we can work it and then something about flipping it and reversing it.

Expect a paper pencil quiz this week and one more tenmarks track that will also count as a quiz.

Transitioning from Polynomial Functions to Exponential Functions begins Groundhog's Day!

Monday Task
Last week we did a couple of open-ended practice task problems.  Monday (if we don't get another foot of snow!), you will complete an open response problem that is probably less difficult than the practice problems we did together, but will still challenge you to make sense of a problem and apply what we have learned to solve problems pertaining to volume in a different context.  This will count as the first test grade of the new quarter.  I don't think most of us really need to do a whole lot of studying for this task, but if you want to spend 10 minutes preparing, it would be a good idea to organize your unit 4 polynomial notes and perhaps log on to www.khanacademy.org and search "Dimensions from volume of box".  There is a short 5-minute video that shows a simpler problem than what you will need to complete, but the core problem solving elements are pretty similar and it should build your confidence.
 Guiding Questions for Tenmarks Problems

To prepare for upcoming material, your weekend homework was to complete 4 www.tenmarks.com tracks to refresh  your memory about the key features and behavior of exponential graphs, structure of exponential equations, and patterns to interpret tables of values modeling exponential functions.

Your first homework grade will be your TM scores for the four assigned tracks.  One student finished all four last Monday in about 30 minutes.  I recommend your complete the tracks and as you are working, answer the following guiding questions:

• Representing and Interpreting Exponential Functions [FIF7e] . For polynomial functions we learned about key features and behavior of that family of function.   Given a simple exponential function y = b^x, what key features do we need to recognize?  What is different about the behavior and rates of change of an exponential function compered to other functions we have studied. 

 

• Understanding Exponential Growth and Decay [F-IF.8b] For polynomial functions, we learned what how each of the parameters in y = a (x-h)^3 +k affects the graph of a polynomial.  What can you recognize about the roles of the a and b for a polynomial function y = ab^x?  What happens if b>1?  What happens when 0<b<1?

 

• Calculating Exponential Growth and Decay Algebraically [F-LE.1c] For linear functions, we know all about positive and negative slope and the y-intercept for any y = mx+b.  Exponentials are not characterized by slope and the y-intercept is not a "b".  What can you generalize about the y-intercept of exponential functions?  What is generally true about the x-intercept(s)?