Cracking the Code: Sinusoidal Equations from Maxima and Minima

Algebra 2 Grades High School 3:26 Video
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Lesson Description

Learn how to derive sinusoidal equations (sine and cosine) when given the maximum and minimum points on a graph. This lesson covers amplitude, period, phase shift, and vertical shift to write the equation.

Video Resource

Finding a Sinusoidal Equation Given a Maximum and Minimum

Mario's Math Tutoring

Duration: 3:26
Watch on YouTube

Key Concepts

  • Amplitude
  • Period and its relationship to the 'b' value
  • Phase Shift (Horizontal Shift)
  • Vertical Shift (Midline)
  • General form of sinusoidal equations (sine and cosine)

Learning Objectives

  • Students will be able to determine the amplitude, period, phase shift, and vertical shift from a given sinusoidal graph's maximum and minimum points.
  • Students will be able to write the equation of a sine or cosine function given its maximum and minimum points.
  • Students will be able to differentiate between using a sine or cosine function based on the graph's starting point.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of sinusoidal equations: y = A cos(B(x - H)) + K and y = A sin(B(x - H)) + K. Define each variable: A (amplitude), B (related to the period), H (horizontal shift), and K (vertical shift). Briefly discuss the relationship between period and B: Period = 2π/B.
  • Example Problem (15 mins)
    Present an example problem where students are given the maximum and minimum points of a sinusoidal graph (as in the video). Guide them through the process of finding each parameter: 1. Amplitude: A = |(Max - Min) / 2| 2. Period: Determine the horizontal distance between the max and min, then double it. Use this value to find B. 3. Phase Shift: Identify the horizontal shift from the standard cosine or sine function. 4. Vertical Shift: K = (Max + Min) / 2 Demonstrate how to plug these values into the general equation.
  • Choosing Sine or Cosine (5 mins)
    Discuss when it might be easier to use a cosine function (when the graph starts at a maximum or minimum) versus a sine function. Note that either function can be used, but one might simplify the phase shift calculation.
  • Practice Problems (15 mins)
    Provide students with additional practice problems where they are given maximum and minimum points and asked to write the corresponding sinusoidal equations. Encourage them to work independently or in small groups.
  • Review and Wrap-up (5 mins)
    Review the key steps and formulas. Answer any remaining questions. Preview upcoming topics related to sinusoidal functions.

Interactive Exercises

  • Graphing Challenge
    Give students several sinusoidal equations and have them graph the functions, identifying the key parameters (amplitude, period, phase shift, vertical shift) on the graph.
  • Equation Matching
    Provide a set of graphs and a set of equations. Have students match each graph to its corresponding equation.

Discussion Questions

  • How does changing the amplitude affect the graph of a sinusoidal function?
  • How does changing the period affect the graph of a sinusoidal function?
  • What is the relationship between the 'B' value and the period of a sinusoidal function?
  • How can you determine the vertical shift from the maximum and minimum values?
  • Why might it be easier to use a cosine function in some cases and a sine function in others?

Skills Developed

  • Problem-solving
  • Analytical thinking
  • Mathematical modeling
  • Graph interpretation

Multiple Choice Questions

Question 1:

The amplitude of a sinusoidal function is calculated by:

Correct Answer: |(Max - Min) / 2|

Question 2:

The vertical shift (midline) of a sinusoidal function is calculated by:

Correct Answer: (Max + Min) / 2

Question 3:

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

Correct Answer: 2

Question 4:

A sinusoidal function has a maximum at y = 7 and a minimum at y = 1. What is the amplitude?

Correct Answer: 3

Question 5:

A sinusoidal function has a maximum at y = 5 and a minimum at y = -3. What is the vertical shift (K)?

Correct Answer: 2

Question 6:

The general form of a cosine function is y = A cos(B(x - H)) + K. What does 'H' represent?

Correct Answer: Phase Shift

Question 7:

The period of a sinusoidal function is determined by the formula:

Correct Answer: 2π / B

Question 8:

Which parameter affects the vertical stretching or compression of the sinusoidal wave?

Correct Answer: Amplitude

Question 9:

Given a maximum at (π/2, 4) and a minimum at (3π/2, -2), what is half the period?

Correct Answer: π

Question 10:

Given a maximum at (π/3, 6) and a minimum at (π, 0), what is the vertical shift?

Correct Answer: 6

Fill in the Blank Questions

Question 1:

The distance from the midline to the maximum (or minimum) of a sinusoidal function is called the _________.

Correct Answer: amplitude

Question 2:

The _________ represents the horizontal shift of a sinusoidal function.

Correct Answer: phase shift

Question 3:

The formula to find the vertical shift (K) given the maximum and minimum values is K = (Max + Min) / _________.

Correct Answer: 2

Question 4:

The period of a sinusoidal function is related to the 'B' value by the formula: Period = 2π / _________.

Correct Answer: b

Question 5:

The _________ shift represents the up and down movement of the sinusoidal function.

Correct Answer: vertical

Question 6:

If a cosine function starts at its _________, then the phase shift might be zero.

Correct Answer: maximum

Question 7:

The general form of a sine function is y = A sin(B(x - H)) + K, where A represents the _________.

Correct Answer: amplitude

Question 8:

To find the period when given a maximum and a minimum, you should determine the horizontal distance between them and then _________ it.

Correct Answer: double

Question 9:

In the sinusoidal equation, y = A cos(B(x - H)) + K, 'K' represents the _________ shift.

Correct Answer: vertical

Question 10:

The general sinusoidal equation uses 'A', 'B', 'H', and 'K' to define its shape and placement on the coordinate plane, where A represents _________.

Correct Answer: amplitude

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together). 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.

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