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Teaching Math: A Video Library, 9-12

Enveloping Functions

Content Standards
Trigonometry, Algebra, Functions, Geometry from an Algebraic Perspective
Process Standards
Reasoning, Communication, Connections
School: Hillside High School; 990 students
Location: Durham, North Carolina
Teacher: Harriette Davis
Years Teaching: 24
Students in Classroom: 20
Grade: 11 – 12
Enveloping Functions
Video Overview
The class reviews graphing in cases when the coefficient of the function is a number and the coefficient is a variable. Students discuss the curve of two functions, and Ms. Davis introduces the concept of boundaries, or envelopes. They consider functions that contain variable factors and graph one function on their calculators. Using the results, students develop conjectures for sets of functions that may provide a boundary function. Students then work in groups to graph a new function. At the end of class, students make generalizations about envelopes for the functions.

The Class Challenge
Students graph the following function:
y = 2^x · sin (x)

Course and Curriculum
Prior to this elective precalculus honors course, students took Algebra 1, Algebra 2, and Geometry. This course includes the study of polynomials, data analysis, functions, discrete topics, and modeling. The graphing calculator is used to discover concepts and make connections within mathematics. This lesson, which concludes a three–week trigonometry unit, was completed the following day, when the students added functions and solved application problems. In the previous unit, students worked on basic graphing and the laws of sines and cosines.
Ms. Davis uses several assessment methods. For example, instead of checking homework each day, she asks students every two weeks to write how they solved five homework problems. At the beginning of each class, students write down the problems they found difficult. The class discusses these problems, and the students who completed them present their work. Presenting students receive extra points. In addition to homework, Ms. Davis' students are graded on class work, such as writing and small group work, and tests.

A Pre–Viewing Exploration for Teacher Workshops
Enveloping Functions
Objective: To investigate products of two functions and explain how the separate factors help to determine the product functions.
Materials: One graphing calculator with the capability to graph functions for each small group.
Activity
Solve this problem in small groups.
Using a graphing calculator, with a range that allows you to see at least four periods of the graph, graph the following functions of the form y = a sin x:
y = (1/2) sin x
y = sin x
y = 2 sin x
y = 4 sin x
Describe how the graphs change as the coefficients change.
Look at the graph of the function y = 2 sin x. Determine the functions that envelop this function. Specifically, what are two functions that contain y = 2 sin x between them in the tightest sense?
Observe the behavior of the function over four periods. Describe how multiplying by 2 influences the graph of y = sin x.
In each case above, the coefficient is a constant. Use a calculator to investigate what changes if the function has a variable coefficient, as in y = x sin x.
Describe what happens in this graph.
What lines envelop this function?
How would you describe the influence of multiplier x on the function y = sin x?
What would change if the coefficient function was exponential instead of linear? Use a calculator to graph y = 2^x sin x and describe what you see.

Exploring Content

If you were given the graphs of two functions, how could you use the graphs to sketch the product function?
What is the difference between envelope functions and asymptotes?
Discuss the symmetry of the functions y = a sin x, y = x sin x, and y = 2^x sin x with respect to the origin, x–axis, and y–axis.

Exploration Answers
Activity

The functions stretch vertically as the amplitudes becomes larger.
The function y = 2 sin x is contained between y = 2 and y = –2.
It causes a constant vertical stretch. The function value on the vertical axis is the product of 2 and sin x.
The product of x and sin x causes the stretch of one graph y = sin x to increase as x increases.
y = x and y = –x
It causes a variable vertical stretch proportional to the value of x.
The stretch growth is exponential and the graph is enveloped by y = 2^x and y = –2^x.

Exploring Content

You could find the product of discrete points for x in each function in order to find the value of the product function at that point.
Envelope functions are tangent to the given curve. When a function is asymptotic to a line called the asymptote, the function's value approaches, but never reaches, the value of the asymptote.
y = a sin x has origin symmetry. y = x sin x and y = 2^x sin x have y–axis symmetry.

Topics for Discussion
How did Ms. Davis build on students' previous knowledge to introduce new ideas?

Discuss how students used geometric and algebraic representations. For example, how did they represent algebraically the geometric idea of a boundary?
How does Ms. Davis help students to organize their thoughts and search for a conjecture?
What are some methods you can use to improve students' understanding and use of mathematical concepts?
How would you move students beyond initial reliance on a calculator–generated graph (inductive reasoning) to a more deductive verification of their conjecture?
How could you make sure you have caught students' misinterpretations of concepts before teaching a new concept?
Discuss techniques for assigning students to groups.

Identify strategies you have used to create student groupings. What are your goals for each strategy?
What is your role in making student group work successful?
How do you group students with different levels of understanding? What other criteria should be considered when grouping students?
Describe different kinds of groupings and the work for which you think they are best suited.
How did Ms. Davis' questions engage students and advance mathematical thinking?

How does Ms. Davis address student responses to ensure that students understand concepts before she introduces another idea?
How can you engage students in reviewing concepts and skills when you teach new concepts?