Unit 6, Lesson 15
Transforming Trigonometric Functions
Integrated Math 3
15.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
The goal of this Warm-up is to recall how a quadratic equation changes when its graph is translated horizontally. This will prepare students for examining this same translation when it is applied to trigonometric functions in this lesson.
Launch
NoneActivity
NoneMatch each equation with its graph. Be prepared to explain your reasoning.
A
B
C
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share how they made their matches. If no students bring up the transformation relationship between the graphs, ask how the graphs are related to each other. (They are the same shape but are horizontally translated.)
Display the three graphs together, and ask students to explain how changing the input in the equation changes the graph. (Adding a positive number to the input translates the graph to the left, and adding a negative number to the input translates the graph to the right.)
15.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
This activity asks students to interpret cosine and sine equations in the context of a fan. They reason abstractly and quantitatively to connect the properties of the fan with the graphs (MP2). They will need to focus on three different aspects of the equations:
- The amplitude of the sine graph (length of the fan blade)
- The midline of the sine graph (height of the center of the fan blades)
- The direction the fan spins (clockwise or counterclockwise).
Launch
Arrange students in groups of 2. Display an image of the blades of a fan, such as this one:
Tell students that these blades extend 4 feet from the center and that the center of the fan is 20 feet off the ground. The fan rotates counterclockwise.
Ask students to write an equation to represent the height in feet of the point, , relative to the ground, as a function of the angle of rotation (). Select 2–3 students to share how they wrote their equation. Tell them that in this activity they are going to work in the opposite direction. That is, they are going to start with an equation giving the vertical position of a point, , at the tip of a fan blade and use it to describe the fan.
MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard, using precise mathematical language and their own words. Display this sentence frame: “I heard you say _____.” Original speakers can agree or clarify for their partner.
Advances: Conversing, Speaking
Advances: Conversing, Speaking
Representation: Access for Perception. Provide students with a physical copy of the image displayed and information provided during the Launch. Check for understanding by inviting students to rephrase the information in their own words. Invite students to label the image based on the information provided. Keep the display of the image and information visible throughout the activity.
Supports accessibility for: Language, Memory
Supports accessibility for: Language, Memory
Lesson Synthesis
The purpose of this discussion is for students to consider a periodic function in which three types of transformations are in effect at once, building on the work from the activities and looking ahead to the next lesson. Display the graph of the function :
Here are some questions for discussion:
- “How does the 2 from the equation show up on the graph?” (2 is the amplitude, the distance from the midline to the maximum value or to the minimum value.)
- “How does the -1 from the equation show up on the graph?” (-1 is the vertical translation or midline, halfway between the maximum and minimum values.)
- “How does the from the equation show up on the graph?” (It translates the graph to the right by .)
If time permits, ask students what non-zero value they could replace with in order for its graph to look the same as the graph of . (Any integer multiple of .)
Student Lesson Summary
The graphs of cosine and sine functions can be translated vertically or horizontally, and the size or height of their graphs can also be modified similarly to how we transformed other types of functions in an earlier unit. Let’s look at the graphs of and .
The coefficient 2 stretches the graph vertically, doubling the amplitude of the sine graph. This means that the distance from the midline to the maximum or minimum value is now 2 instead of 1. Adding 3 to the equation translates the midline up by 3 units.
What if we want to translate the graph of to the left? We can do this by adding an angle to the input, . Let’s look at the graph of . The graph of this function looks just like the graph of translated to the left by . (It also looks a lot like !)