Transformations of Sine and Cosine (Lesson 6.6)
Unit 6 Day 8
CED Topic(s): 3.6
Determine how the amplitude, period, domain, range, and midline of sinusoidal functions are affected by transformations.
Graph transformed sine and cosine functions given an equation.
Quick Lesson Plan
In this lesson students will use the “Which One Doesn’t Belong?” structure to explore how transformations affect the parent functions of sine and cosine. Although students are used to seeing horizontal and vertical shifts, stretches, and shrinks, they are not used to thinking about how these transformations affect the amplitude and period, as these are new concepts for periodic functions.
Although the original “Which One Doesn’t Belong?” structure has students only pick one item that doesn’t belong, it’s important that students are able to provide a convincing argument for each graph, as each one highlights a different transformation. Students first think visually about how the graph differs, and then in question 2 are asked to reason about how these differences can be seen in an equation. As always, we want students connecting concepts across multiple representations. In question 3, we return to the important idea of equivalence, a thread that has been woven through the entire course.
As time permits, have groups share and critique each other’s convincing arguments. As you monitor students working on the activity, be looking for groups that can present unique perspectives on the differences between the four graphs and equations. In the discussion, you may want to begin with the graph of amplitude 2 (Graph A) and end with the graph that has a period of π (Graph C) as this observation tends to be the most challenging for students to articulate.
Does a vertical stretch by a factor of 2 have the same effect as a vertical shift up 1? Why or why not?
What kind of transformation occurred here? Was it a rigid transformation or was it a dilation?
Which of these graphs has a different domain than the parent function?
Which of these graphs has a different range than the parent function?
Can you think of another equation that would have the same graph as y=sin(x-π)?
As much as possible link the language of vertical stretches/shrinks with changes in the amplitude and horizontal stretches/shrinks with changes in the period. Instead of simply having students memorize the formula for how to find the period of a trig function, ask students to first consider whether the graph has been compressed or expanded and if this means the period has gotten smaller or bigger. Have students notice patterns between the value of the coefficient of the independent variable and the period of the graph. Students should recall that a horizontal stretch of a factor bigger than 1 actually shrinks the graph (meaning compression) which would shorten the period, just as dividing by value bigger than 1, decreases the value.
In our classes we use the phrases “horizontal shift” and “phase shift” interchangeably so that students are familiar with both terms.
Be careful when writing equations that include both a phase shift and horizontal dilation as these are particularly challenging for students. At least at the beginning of this topic, we avoid questions that include both, or we give the equation as sin (k(x-c)) where the dilation factor is factored out.