When class begins, I will project the graph of the voltage equation given in the AC Generator Problem using GeoGebra or a similar graphing application. My students began this problem in the previous lesson, and it's time to draw some conclusions from their progress so far.
First, we'll adjust the axes to show three complete cycles, and we'll take a moment to identify the roots of the function, as well as the maxima and minima. For this, I'll act as the scribe for the students as they tell me what to do. "You need to make your V axis go up to 220 and down to -220. ... Your t axis should only go to 1/12th of a second. ... Um, that's about 0.083 seconds." Before long, we should have a useful graph to work from. I want to go through this process with them to reinforce the notion that the appearance of the graph is highly dependent on the scaling of the axes! (MP 5)
Next, I'll call on individual students (the quiet ones) to explain their solutions to parts b, c, and d. For each one, I'll ask them to point out how the graph illustrates it, and I'll also ask them if they can "see" the solution inherent in the equation. We won't make much of this quite yet, but I want them to notice the "coincidence" that 220 is the maximum and 220 is the coefficient of the sine function.
Finally, I'll call on someone to come to the board to explain how they obtained an equation for the current in part e. They should begin with Ohm's Law, substitute for V and R, and then solve for I.
Now that we have a new sine function, I would like to encourage my students to make some predictions about the graph based on the structure of the function (MP 7).
First, notice the similarity between the two function - the only difference is in the coefficient!
Great, now we can scale both of our axes and make a graph, and it should go more quickly this time!
For the next 15 minutes, students will work individually or in small groups to complete the remaining parts of the AC Generator Problem.
The primary challenge at this point will come in the final part. To answer this question, students will need to use the inverse sine function and correctly interpret the results in context. This is not easy, but provides a healthy challenge for my best students. What is most important is that everyone completely understand parts f, g, and h.
An alternative to the inverse sine function is to make use of the fact that sin(x) is approximately equal to x for small values of x. The class saw that this was true in a previous lesson, and they could put that knowledge to use here. Their answer would be slightly inaccurate, but in a modeling context like this, the error may be acceptable. (MP 4)
Finally, the interpretation of the resulting t-value requires students to make use of the symmetry of the sine function. Essentially, what they are finding is the change in t that corresponds to an change in V from 0 to +/- 8.
Using GeoGebra (you can download the file in the resources), I'll project a graph of y = sin(x) for all to see.
"Based on what you've seen, how can we make the graph oscillate more or less than 1 unit from its axis/midline?"
Students should be able to explain that multiplication is the key. Since the maximum of sin(x) is 1, then the maximum of A*sin(x) must be A. We will consider several values of A with GeoGebra. This leads us to the general equation: y = A*sin(x). I'll make a note that A = amplitude.
"Is it possible to shift this graph upward or downward? I'd like it to oscillate around the line y = 3."
Although they did not experience this with the AC Generator problem, I expect it to be fairly simple based on their experience with other function types. This leads to the general equation: y = C + A*sin(x). I'll make a note on the board that C = vertical shift.
"This graph completes one cycle every 2 pi units on the x-axis. How can we make it complete more cycles for the same change in x?"
Another way to ask this is, "Is there any way to 'trick' the function into thinking that it should end the cycle at an arbitrary x-value?" This one is quite a bit more challenging. Based on the AC Generator problem, however, students will know that they must multiply x by some factor before taking the sine of the argument. The trick is to be able to explain why and I will give my students the opportunity to try doing so. This will lead to careful definitions of the terms cycle, period, and frequency and to the general equation: y = C + A*sin(Bx). I'll note that B = frequency.
[The CCSS does not require students to consider phase shift at this point. I won't avoid the question if it comes up, but I won't include it in the general equation at this point.]
When we're all done, I like to take a step back and point out what we've accomplished. Not only have we redefined "sine" in terms of the unit circle, but we've turned it into a function capable of being evaluated for all real numbers and whose graph makes perfectly regular oscillations! That's quite an accomplishment - and it will be extremely useful!