Lesson: Exponential Functions (non-linear functions)

"Hole puncher, where are you!? It usually comes to me when I call it."
- Mr. Max :)

We started with Mental Math. #17

Things from mental math:

x - linear (no curve)

x^2 - quadratic (one curve)

x^3 - cubic (two curves)

x^4 - quartic (three curves)

x^5 - quintic (four curves)

Pascals Triangle

- All Hallow's Day/All Saints Day (Nov 1) is after All Hallow's Eve (Oct 31)

- 7 months have 31 days

Homework Check for pg A-7 to A-9

25 objectives due thursday at 3:40

Exponential Functions

This slide is the summary notes for the lesson (exponential functions will always be in the form y=ab^x), and how exponential functions should generally look (rabbits multiply quickly over a certain period of time).

This slide is of your first example (hope you can read the example). We did two examples today. This one is about the intensity of light at a certain distance below the surface of a pond.

"You are a little frog sitting on a lily pad basking in the August sun. Something happens that the lily pad is pulled out of underneath you, and you find yourself a meter under the surface of the water."

The point of that was to 'illustrate' how the intensity of the light is less as you go deeper into the pond. If you were pulled a meter under the water you still see a lot of light around you because only 3% of the light was lost. We caught on quickly to the obvious - the intensity of light at the surface of the water is at 100%.

With this information we came up with our equation based on the question (get my drift?). We came up with I(d) = 100(0.97)^d. We chose this equation (out of the possible two/four) because it is a.
Where the 0.97 is the percentage of sunlight you see one meter below the surface of the water.
With that equation we entered it into our calculator/graphed it (substituting I(d) for y and d for x), and got something very similar to what Max has on the slide. We determined that a was the correct answer after observing the graph (at 0 meters below the surface we received 100% of the sunlight, at one meter 97% of the sunlight was showing through, at two meters we were able to see 94.09% of sun rays).

This slide is the second example. I'll 'retype' the question since you can hardly see it in the picture.
"Early in the 1900s, an airplane manufacturer was able to increase the time its airplane could stay aloft by constantly refining its techniques. Using a graphing calculator, determine which exponential equation best models the data."
With this question we are given a table with 'years after 1900' and 'time aloft'.
With the green on the slide Max isolated exponential equation and wrote "do regression!", so that's what we are going to do (with our calculators... in point form because this post is beginning to grow like the bunnies).

- Determine which piece of information/variable is dependent (y) and independent (x). In this case 'time aloft' would be dependent so it will be labeled L2. (likewise the independent variable would be L1).
- On you calculator enter your information in the stat plot (If you don't know how/forgot go to 'the old math blog' or :chevy:'s post... apparently. This is also the top left clip on the slide). You might want to turn on that stat plot also.
- After you have your plot you could use zoom 9 or the window settings that are on another clip on the slide above. There you have the plot (right above the window settings).
- To draw the exponential regression (trying hard not to call it linear regression), go to STAT, CALC, 0:ExpReg.
- You end up back at your home screen. Press L1, 'comma', L2, 'comma'. To get the Y1 (as illustrated on the slide) go to VARS, Y-VARS, 1: Function, 1: Y1, Enter.
- ENTER, and you have your a and b variables which should tell you your answer.
- To see the regression simply go back to your graph and 'see' it.

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All info. came from inside the classroom and out of my keyboard (accept for pascals triangle which is a link in itself).
A point of interest... you can never have zero light. There will never be perfect and complete darkness.
People seemed to think my cartoon on my last post was pretty good, I'll give yea another one (which will likely end up as my signature with a smiley face just below it).

Which variable is dependent and which is independent? Where is the title for that matter?

:)