Shifting X and Y Values
The way I taught my students, this semester, was extremely beneficial for all the horizontal shifts that occur in terms of period changes and horizontal/phase shifts. The way I teach my kids is to attack the graph in a few steps.
1) Graph the original function.
2) Relabel your y-coordinates by first checking for a vertical shift. This will tell you if you are "centered" around the y=0 line. If there is a vertical shift. Change this from 0 to whatever you shift is.
-Next, attack the amplitude by relabeling your maximum and your minimum. Add your amplitude to this vertical shift and relabel this max. Subtract your amplitude from your vertical shift and relabel your minimum.
3) Next came the hard part. There are "two rounds" of relabeling the x-coordinates. The first round, we use the new period. This way, we have then "started" at x = 0. Relabel your end of the first cycle with the new period. I then broke this up into chunks using 1/4, 1/2, and 3/4 of the period. Do this to both sides.
4) The "second round" involves the phase shift. Add or subtract the phase shift to each of your x-coordinates. This was really hard for my kids as I learned that fractions were not their strong suits. They hated me even more as I did not allow a calculator.
That is right, I took away the calculator for graphing trig functions. MEAN MISS RUDOLPH!!!
I will say, a student told me, after he had already passed my class: "Miss Rudolph, your class and unit 5 made me much better at fractions. I actually understand them now because I was forced to learn them."
That right there, is why I loved the way I taught this.
Of course, steps 3 and 4 can be condensed once the kids have understood that you are shifting the graph by doing two rounds. Eventually, I started by making the phase shift at x=0 and then figuring out what the mark would be at the end of the cycle so I had one full period, based upon whatever this was.
I've included an image of one of the homework problems where I showed all my steps for my kids so perhaps you too can figure out what all this process was!
Pros: No Calculator!!!! Fraction Skills Improved, Learned Basic Shapes of Functions
Cons: I'm not sure everyone fully grasped what this would look like on a calculator.
What are some other ways that you teach graphing trig functions?