Unlocking Sinusoidal Equations: Sine vs. Cosine

Algebra 2 Grades High School 6:20 Video
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Lesson Description

Learn to write equations for sinusoidal graphs using both sine and cosine functions. Explore amplitude, period, phase shifts, and vertical shifts to represent the same graph in multiple ways.

Video Resource

Sine or Cosine Writing Equations Given Graph

Mario's Math Tutoring

Duration: 6:20
Watch on YouTube

Key Concepts

  • Sinusoidal functions (sine and cosine)
  • Amplitude, period, vertical shift, and phase shift
  • Transformations of functions (reflection, shifts, stretches/compressions)

Learning Objectives

  • Students will be able to identify the midline, amplitude, and period of a sinusoidal graph.
  • Students will be able to write equations for a given sinusoidal graph using both sine and cosine functions.
  • Students will be able to determine the necessary phase shift and vertical shift for different representations of the same graph.

Educator Instructions

  • Introduction (5 mins)
    Begin by briefly reviewing the basic shapes of sine and cosine graphs. Explain that the lesson will focus on writing equations for sinusoidal graphs, highlighting the flexibility of using either sine or cosine.
  • Video Explanation (10 mins)
    Play the video 'Sine or Cosine Writing Equations Given Graph' by Mario's Math Tutoring. Encourage students to take notes on the key steps: finding the midline, amplitude, period, and determining the phase shift. Pause at key moments (e.g., 1:10, 2:58, 3:37, 4:56) to emphasize important concepts.
  • Midline, Amplitude, and Period (10 mins)
    Discuss how to identify the midline, amplitude, and period from a sinusoidal graph. Emphasize that the midline represents the vertical shift (K value), the amplitude is the distance from the midline to the maximum or minimum, and the period is the length of one complete cycle.
  • Sine vs. Cosine: Choosing a Starting Point (10 mins)
    Explain that the choice between sine and cosine depends on where you choose to consider the "starting point" of the graph. Cosine naturally starts at a maximum or minimum, while sine starts at the midline. Discuss how reflections (negative A value) affect the choice of function and phase shift.
  • Practice Problems (15 mins)
    Provide students with several sinusoidal graphs and have them write equations using both sine and cosine functions. Encourage them to explore different phase shifts and reflections to represent the graph in multiple ways. Have them work independently and then share their solutions with a partner.

Interactive Exercises

  • Graph Matching Game
    Present students with a set of sinusoidal graphs and a set of equations. Have them match the graphs to their corresponding equations. Include multiple equations for each graph to emphasize the different possible representations.
  • Equation Challenge
    Give each student a different sinusoidal graph and challenge them to find as many different equations as possible to represent it. Encourage creativity and collaboration.

Discussion Questions

  • Why can the same sinusoidal graph be represented by multiple different equations?
  • How does a negative amplitude (reflection) affect the phase shift needed to write the equation?
  • What are the advantages and disadvantages of using a sine versus a cosine function to represent a given graph?

Skills Developed

  • Graph analysis
  • Equation writing
  • Problem-solving
  • Critical thinking

Multiple Choice Questions

Question 1:

The ________ of a sinusoidal function is the distance from the midline to the maximum or minimum point.

Correct Answer: Amplitude

Question 2:

What does the 'B' value in the equation y = A sin(Bx - C) + D affect?

Correct Answer: Period

Question 3:

Which of the following transformations does a negative 'A' value (e.g., y = -A sin(x)) represent?

Correct Answer: Reflection over the x-axis

Question 4:

The midline of a sinusoidal graph represents the:

Correct Answer: Vertical Shift

Question 5:

A phase shift in a sinusoidal function results in a:

Correct Answer: Horizontal translation

Question 6:

If a sinusoidal graph starts at its maximum value, it can be easily represented by a ________ function.

Correct Answer: Cosine

Question 7:

The formula to calculate the period of a sinusoidal function is:

Correct Answer: Period = 2π / B

Question 8:

Which transformation is represented by the 'D' value in the equation y = A sin(Bx - C) + D?

Correct Answer: Vertical Shift

Question 9:

Which of the following is the parent function for cosine?

Correct Answer: y = cos(x)

Question 10:

Which of the following is the parent function for sine?

Correct Answer: y = sin(x)

Fill in the Blank Questions

Question 1:

The line that splits a sinusoidal graph in half is called the ________.

Correct Answer: midline

Question 2:

The ________ is the horizontal distance required for a sinusoidal function to complete one full cycle.

Correct Answer: period

Question 3:

A ________ shift moves a sinusoidal graph horizontally.

Correct Answer: phase

Question 4:

If the 'A' value in y = A sin(x) is negative, the graph is ________ over the x-axis.

Correct Answer: reflected

Question 5:

The general form of a sinusoidal equation is y = A sin(Bx - C) + D, where D represents the ________ shift.

Correct Answer: vertical

Question 6:

When writing an equation for a sine function that starts at a maximum, it often involves a ________ shift compared to the basic sine function.

Correct Answer: phase

Question 7:

The ________ of a wave can be calculated by finding half the distance between the max and min values of a graph.

Correct Answer: amplitude

Question 8:

The standard period for both parent sine and parent cosine is _______.

Correct Answer: 2π

Question 9:

In the equation y = A*cos(Bx), the ____ effects the period.

Correct Answer: B

Question 10:

In the sinusoidal equation, y = A sin(Bx - C) + D, 'A' stands for ________.

Correct Answer: amplitude

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.1.5:Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease.

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