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Unit 7, Lesson 14

Amplitude and Midline

Algebra 2

14.1

Warm-up

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

The goal of this Warm-up is for students to recall the effects of vertical translations and stretches in the familiar setting of quadratic functions, before considering these transformations' effects on trigonometric functions.

Launch

None

Activity

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Match each equation to its graph. Be prepared to explain how you know which graph belongs with each equation.

<p>parabola opening up with vertex at orgin. narrower than the graph of y = x squared.</p> — A

<p>parabola opening up with vertex at origin.</p> — B

<p>parabola, opening up with x intercepts of -1 and 1 and vertex at -1 comma 0. </p> — C

<p>parabola opening up with vertex at 0 comma -1, x intercepts between -1 and 0 and between 0 and 1.</p> — D

<p>parabola opening up, with vertex at 1 comma 0.</p> — E

Student Response

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Building on Student Thinking

Activity Synthesis

Invite students to explain their matches for each of the equations. Encourage them to refine their descriptions of the transformations of , using more precise language and mathematical terms (MP6), and connect this matching activity to the work done in a previous unit on transformations of functions.

Ask students, “What other transformations can you think of using the values 1 and 3, starting with the function ?” (, whose graph is stretched vertically by a factor of 3 and then translated up 1 unit. Or, , whose graph is translated to the left by 3 units.)

14.2

Activity

Instructional Routines

Aspects of Mathematical Modeling Graph It

Materials

To Gather

Graphing technology

Activity Narrative

The goal of this task is to introduce the amplitude of a trigonometric function in the context of the position of a point on a spinning windmill. In this case, the amplitude of that position depends on the distance of the point from the center of the windmill. The closer the point is to the center, the smaller the amplitude of the vertical and horizontal positions of the point. The general shape of the graph is the same, rising and falling as the windmill blade makes a complete revolution; what changes is how high and low the graph goes. This is analogous to how the 3 in changes the shape of its graph compared to the graph of ; it stretches the graph vertically so the output increases more quickly, from 0 to 3 instead of from 0 to 1, when goes from 0 to 1.

As students determine what type of model best describes the situation, adjust the trigonometric function to account for the amplitude, and check their predictions using technology, students are engaging in aspects of mathematical modeling (MP4).

14.3

Activity

Instructional Routines

Graph It MLR8: Discussion Supports

Materials

To Gather

Graphing technology

Activity Narrative

The goal of this activity is to introduce the midline of a trigonometric function. Students experiment with changing the vertical position of a trigonometric function, adding a constant within the same context of the height of a windmill. Unlike the scalar multiple in the previous activity, a vertical translation is a rigid transformation and so does not change the shape of the graph, just its location. Students reason quantitatively and abstractly as they consider the effects of changes to the position and shape of the windmill blades on the features of the graph (MP2).

Launch

Graphing technology is needed for every student.

Representation: Access for Perception. Read the information about the windmill aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language, Attention

Lesson Synthesis

The purpose of this discussion is for students to consider a periodic function for which both the amplitude and the midline are not 1. Display a graph of :

<p>A graph of 2 cosine theta minus 3. </p>

Here are some questions for discussion:

“What is the midline of ? Explain how you know.” (. It is halfway between the highest value, -1, and the lowest value, -5.)
“What is the amplitude of ? Explain how you know.” (2. The coefficient of is 2, so the graph has a vertical stretch of 2, which is the amplitude. Or, the maximum and minimum values of the graph are 2 away from the midline value.)

Draw the midline onto the graph, and indicate where to see the amplitude on the graph, indicated by the length of the dashed vertical line shown here:

<p>A graph of 2 cosine theta minus three with midline labeled. </p>

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Student Lesson Summary

Suppose a bike wheel has a radius of 1 foot and we want to determine the height of a point, , on the wheel as it spins in a counterclockwise direction. The height, , in feet of point can be modeled by the equation , where is the angle of rotation of the wheel. As the wheel spins in a counterclockwise direction, the point first reaches a maximum height of 2 feet when it is at the top of the wheel, and then a minimum height of 0 feet when it is at the bottom.

<p>A circle, subdivided by 12 congruent central angles. Point P is on the circle on the right side of the horizontal diagonal. Radius is 1 foot.</p>

The graph of the height of looks just like the graph of the sine function but it has been raised by 1 unit:

The horizontal line , shown here as a dashed line, is called the midline of the graph.

<p>graph of sine function with midline at the horizontal line h = 1. </p>

What if the wheel had a radius of 11 inches instead? How would that affect the height, , in inches, of point over time?

This wheel can also be modeled by a sine function, , where is the angle of rotation of the wheel. The graph of this function has the same wavelike shape as the sine function, but its midline is at and its amplitude is different:

The amplitude of the function is the length from the midline to the maximum value, shown here with a dashed line, or, since they are the same, the length from the minimum value to the midline. For the graph of , the midline value is 11 and the maximum is 22. This means that the amplitude is 11 since .

<p>graph of sine function with midline at the horizontal line h = 11 and amplitude = 11. </p>

For the graph of , the midline value is 11 and the maximum is 22. This means that the amplitude is 11 since .

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Lesson 14

Amplitude and Midline

Warm-up

Standards Alignment

Building On

HSF-BF.B.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

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Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

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Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Addressing

Building Toward

Standards Alignment

Building On

HSF-BF.A.1

HSF-BF.B.3

Addressing

HSF-IF.C.7.e

HSF-TF.B.5

Building Toward

Standards Alignment

Building On

HSF-BF.A.1

HSF-BF.B.3

Addressing

HSF-IF.C.7.e

HSF-TF.B.5

Building Toward