Today we start by sharing our answers to the assignment from yesterday. The homework assignment was to list the transformations for the functions f(x) = cos (2x - 60°) and g(x) = cos (2(x-60°)) and to try to graph the two functions. A common occurrence is that students will graph both of these functions with a phase shift of 60°. I want to address this issue as soon as possible so students can immediately get in the habit of factoring out the coefficient of x when graphing.
We will start the discussion by asking students to present what they thought the transformations were before they actually graphed them. Students may say that there should be a horizontal shift of 60° to the right for both functions, or some may say that f(x) will have a horizontal shift of 30°. I would use Desmos to show both functions on the same graph so that students can see that they are different – that extra set of parentheses does make a difference! Now we can talk about the actual phase shift for each function. In the video below I give a strategy to find the phase shift of the function.
When discussing the coefficient of the x, I often say it is like the speed of the function. In this case we are going twice as fast so instead of it taking 360° to travel one full cycle, it will take us 180°.
Now that we have conquered those issues, we can add in some more transformations to get students comfortable with sketching graphs. Again, my students graphed these functions last year, so I am giving a quick overview of these topics before we move on to trigonometry concepts that are brand new. On slide 2 of the PowerPoint, students will graph transformations of a sine function. I have sequenced the transformations so that we are adding one transformation each time with graphing increasing in complexity.
This is a great strategy because you can pinpoint exactly which transformation will give students a tough time. Also, students will know exactly which transformation is confusing to them; it is a good tool for a self-assessment of what they remember from Algebra 2. I structured these transformations in this way because this is an easy way to graph when you are given all transformations at once. First you graph the vertical shift to get the new midline, then the horizontal shift gives you the starting value of the function, then the amplitude dictates how high and how low the graph goes, and then the period tells you where to end your cycle.
Choose a group of students to present their graphs to the class. Have them discuss the transformation and say how they can identify the transformation by looking at the equation. Then have them discuss how it affects the graph. You want students to fluently be able to go from equation to graph.
On slide 3, students have a graph that includes all of the transformations. Give them this one and see if they can identify the structure from slide 2 and think about how the individual transformations can be gone through step by step to get the final graph. This is something that you will definitely address if it does not come up.
Since slide 2 involves radians, it can be a little bit tricky to find the period of this function. If a student gets stuck, direct them back to the first examples to see how we got the period of the function by using the “speed.” This should hopefully signal the need to divide the original period (2pi radians) by the “speed” (pi/4 in this case).
Here is an assignment to extend the work that we did in class today. In addition to graphing transformations, students will be given a graph and will be writing the equation of the function. There is also some more work with the Ferris wheel scenario that we worked on earlier in the unit.