**Learning Goals**:

Investigate the exponential function using a finite difference table

Investigate the inverse of the exponential function using graphical methods and describe its key features

**Text reference**: 6.1 (see attached)

**Day 1**__Quick Review__: The work we did last week was review from grade 11, hopefully more rigorous. Ask if you have any remaining worries about this material. Note that drawing functions accurately by hand is a great way to understand transformations. Plotting all of them on the TI84 is not that instructive but is good for checking your answers.

__Finite Differences__: see pp. 310-313 in the textbook and do Investigation #1

Use your TI84 to Part A. If you remember how to lock and edit lists you can look at a lot of examples quickly but you can do it by hand if you prefer. The point is to understand the ideas.

After you have answered the Reflect question (#6), look at the attached Excel file (6.1 Finite Difference Table) to see a much larger table of differences for y=b^x. Predict the values in the table for a variety of bases and check if you are correct. Note that there is more than one pattern and the second one is subtle. See if you can identify it but you may not be able to predict it without some Calculus (Something to look forward to!)

Use this GeoGebra file for Part B

https://www.geogebra.org/m/xbrq35ud (Look at it on a full screen.)

You could make your own GeoGebra model to answer #9 or you could base your answer on the table of finite differences in Part A. Why this works will not be easy to explain here.

Exercise - do the Communicate Your Understanding question C1 on p. 318.

Explain how you can recognize whether or not a function is exponential by examining its a) finite differences, b) graph.

You should also be able to explain your answers to the Reflect questions in the Investigation.

**Day 2**__The Inverse of the Exponential Function__: What is an inverse function? There are several ways to explain this using words, graphs, a bubble diagram or by explaining how to find the inverse algebraically. If you can’t remember from gr 11, look at

https://en.wikipedia.org/wiki/Inverse_function read the definition and look at the first diagram. (Don’t read the rest unless you are bored in which case it is quite interesting.)

Look at this GeoGebra animation and explore the graph of the inverse of y=b^x

https://www.geogebra.org/m/avnnn8q4 (Look at if full screen)

Watch the Courseware video Logarithmic Functions Parts 1,2,3.

https://courseware.cemc.uwaterloo.ca/8/35/assignments/208/0 Do Review 01,02,03 and Practice Exercises 1-5. Check your answers

**Day 3** https://courseware.cemc.uwaterloo.ca/8/35/assignments/208/0 Look at Investigate 1. Watch the first part of lesson Part 4, up to Transformations of Logarithmic functions. (We will do this later.)

Do Review 04.

Do the assessment questions by yourself without looking at any resources and email a scan to your teacher by 3pm. The

**questions will be posted Thursday morning** and the answers will be posted on Friday.

__Semi-log Graph paper__: Watch the video

https://www.youtube.com/watch?v=70UT_GA-GTg and then read the page

https://msu.edu/course/fsc/441/semilabel.html Download and print a copy of the semilog.gif and plot graphs of y = 10^x, y=2^x and y=5^x. Explain the characteristics of these graphs.

**Assessment for the Week**: Email your assessment questions and your Learning Log to your teacher by the end of the day on Thursday April 2.