Transforming Trigonometry: Graphing Sine and Cosine Functions

Algebra 2 Grades High School 14:07 Video
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Lesson Description

Master the art of graphing sine and cosine functions with transformations! This lesson covers amplitude, period, phase shifts, and vertical shifts through multiple examples.

Video Resource

Graphing Sine and Cosine Functions with Transformations (Multiple Examples)

Mario's Math Tutoring

Duration: 14:07
Watch on YouTube

Key Concepts

  • Amplitude
  • Period and Frequency
  • Phase Shift (Horizontal Shift)
  • Vertical Shift
  • Reflection over the x-axis
  • Unit Circle Relationship to Sine and Cosine

Learning Objectives

  • Students will be able to identify the amplitude, period, phase shift, and vertical shift from a sine or cosine function's equation.
  • Students will be able to graph sine and cosine functions with various transformations.
  • Students will be able to determine the equation of a sine or cosine graph given the transformations.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic sine and cosine graphs using the unit circle. Briefly discuss how the y-coordinate on the unit circle relates to the sine function and the x-coordinate relates to the cosine function for angles from 0 to 2π. Show the basic shape of the sine and cosine curve.
  • Amplitude and Period (10 mins)
    Explain the concept of amplitude as the vertical stretch or compression of the graph. Introduce the formula for the period: 2π/B, where B is the coefficient of x. Work through an example, calculating the period and amplitude and demonstrating its effect on the graph.
  • Phase Shift and Vertical Shift (15 mins)
    Explain the concepts of phase shift (horizontal shift) and vertical shift. Emphasize that the phase shift is the *opposite* of what the equation might suggest (e.g., (x - π/4) shifts the graph to the *right* by π/4). Explain that vertical shifts move the 'midline' of the graph. Work through several examples that incorporate both phase and vertical shifts.
  • Reflection over the X-axis (5 mins)
    Explain that a negative sign in front of the sine or cosine function reflects the graph over the x-axis. Show an example of how the basic shape changes when reflected.
  • Complex Examples and Graphing Strategy (15 mins)
    Work through multiple examples combining all transformations. Emphasize a step-by-step approach: 1) Identify amplitude, period, phase shift, and vertical shift. 2) Determine the scale of the x-axis by dividing the period by 4. 3) Plot the 'starting point' based on phase and vertical shift. 4) Sketch the basic sine or cosine shape, adjusted for amplitude and reflection. 5) Refine the graph with accurate scaling.
  • Practice and Review (10 mins)
    Provide students with practice problems to work on individually or in small groups. Circulate to provide assistance and answer questions. Review key concepts and address any remaining areas of confusion.

Interactive Exercises

  • Graphing Challenge
    Provide students with an equation of a transformed sine or cosine function. Have them work in pairs to identify the transformations and graph the function. The pair then presents their graph to the class.
  • Equation Match
    Create cards with graphs of transformed sine and cosine functions and separate cards with the corresponding equations. Students match the graphs to the correct equations.

Discussion Questions

  • How does changing the amplitude affect the range of the function?
  • Why does the phase shift appear to be the 'opposite' of the sign in the equation?
  • How can you quickly identify the period of a transformed sine or cosine function?
  • Describe the relationship between the unit circle and the graph of sin(x) and cos(x).

Skills Developed

  • Analytical Skills
  • Problem-Solving
  • Visual Representation
  • Mathematical Communication

Multiple Choice Questions

Question 1:

What does the 'A' represent in the general form of a sinusoidal function y = A sin(Bx - C) + D?

Correct Answer: Amplitude

Question 2:

How does the graph of y = cos(x) differ from the graph of y = sin(x)?

Correct Answer: It is shifted horizontally.

Question 3:

The period of the function y = sin(2x) is:

Correct Answer: π

Question 4:

Which transformation does the '+ D' represent in the general form of a sinusoidal function y = A sin(Bx - C) + D?

Correct Answer: Vertical Shift

Question 5:

What is the phase shift of the function y = cos(x - π/2)?

Correct Answer: π/2 to the right

Question 6:

Which of the following transformations will result in a reflection of the graph over the x-axis?

Correct Answer: y = -sin(x)

Question 7:

The graph of y = 3sin(x) has an amplitude of:

Correct Answer: 3

Question 8:

If the period of a sinusoidal function is 4π, what is the value of 'B' in the equation y = sin(Bx)?

Correct Answer: 1/2

Question 9:

Which of the following equations represents a sinusoidal function shifted vertically upwards by 5 units?

Correct Answer: y = sin(x) + 5

Question 10:

The scale on the x-axis is determined by dividing the ______ by 4.

Correct Answer: Period

Fill in the Blank Questions

Question 1:

The vertical stretch or compression of a sine or cosine graph is called the ________.

Correct Answer: amplitude

Question 2:

The horizontal shift of a sine or cosine graph is called the ________.

Correct Answer: phase shift

Question 3:

The distance it takes for a sine or cosine function to complete one full cycle is called the ________.

Correct Answer: period

Question 4:

A negative sign in front of the sine or cosine function causes a ________ over the x-axis.

Correct Answer: reflection

Question 5:

The formula to calculate the period of a sinusoidal function is 2π divided by ________.

Correct Answer: B

Question 6:

A vertical shift moves the ________ of the graph up or down.

Correct Answer: midline

Question 7:

The graph of y = cos(x) starts at its ________ value.

Correct Answer: maximum

Question 8:

The graph of y = sin(x) starts at its ________ value.

Correct Answer: midline

Question 9:

To determine the phase shift from an equation, remember that the shift is the ________ of the sign in the equation.

Correct Answer: opposite

Question 10:

Factoring out a constant from the argument of a trigonometric function is crucial for correctly identifying the ________.

Correct Answer: phase shift

Educational Standards

A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically. A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain.

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