The first thing was a comment from a student after I gave back a test (which, by the way, no one studied for! They admit it at least, but apparently the juniors need to read the article on how to study math too.) There was a question asking students to compare y=sec(x) and y=sec(2x)-3. One student said to me "I feel like I just have to memorize all these parts that don't make sense." At which point I thought "Um, yes, but also no" We have done a lot of examples, made connections and they had use of a graphing calculator to answer that question so it wasn't just about memorizing. But it was also concerning that this student hadn't figured the role of each piece out yet since we've been talking about them ad nauseum.
Then I went to a class this evening at the EDC and the awesome Bowen Kerins shared an insight about shifting and stretching functions. First: the whole "inside the function is backwards" doesn't sit well with him or with students (or with me) and, all of these rules are null when you get to something like a circle where there is no "inside the function." Instead, he suggests using a change of variable to get back to the parent function. The strategy of chunking is useful all over the place, and by the time students get to calculus, u substitution will seem obvious. So here's how it works:
y=(x-5)2 - 7
Make a table for M, N (so much easier since the vertex is at 0,0).
M=x-5 x=M+5 (Shift right 5)
N=y+7 y=N-7 (Shift down 7)
To find the point you plot, use a fancy tool some call a pointer finger: cover the M and N, plot the x and y. The shifts are right there in the linear transformations you did, and they always work, no inside, outside, intuitive, counterintuitive or solving to find the vertex. It makes sense! Now if only I'd known this before all this work on shifting and stretching.
And, it works for any type of function, let's try a sine wave:
y - 4=sin(πx+π)
Even though the period of this function is 2 and the usual suspects for x values will give ugly y values, the usual suspects work perfectly for M and N.
M=πx+π x=(M-π)/π (Shift left π, shrink/change the period by dividing by π)
N=y - 4 y=N+4 (Shift up 4)
My students always struggle with picking the "right" x and y values to get the real shape of the graph, including the maximum and minimum. With this method there's no need to figure out good x values to choose, they just need to know the parent function well enough and learn that one set of "good x values."
P.S. I left for school at 7 am, ran a club from 2-3 and was in class from 4-8, 45 minutes away. Over twelve hours out of the house and I still blogged! I didn't proofread, but I shared an idea. I will fill in the table and maybe another example tomorrow.
*Edited 11/6 to add tables and trig example