Decoding Sinusoidal Equations: From Maxima and Minima to Graphs

PreAlgebra Grades High School 5:21 Video
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Lesson Description

Learn to derive sinusoidal equations (sine and cosine) from given maximum and minimum points, mastering amplitude, period, phase shift, and vertical shift.

Video Resource

Writing a Sinusoidal Equation Given Max and Min (Example 2)

Mario's Math Tutoring

Duration: 5:21
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Key Concepts

  • Amplitude
  • Period
  • Midline (Vertical Shift)
  • Phase Shift (Horizontal Shift)
  • Sinusoidal Functions (Sine and Cosine)

Learning Objectives

  • Determine the amplitude, period, midline, and phase shift from given maximum and minimum points of a sinusoidal function.
  • Write both sine and cosine equations that model a sinusoidal graph, given its maximum and minimum points.
  • Understand the relationship between sine and cosine graphs as horizontal shifts of each other.

Educator Instructions

  • Introduction (5 mins)
    Briefly review sine and cosine graphs, their properties (amplitude, period), and the general forms of sinusoidal equations. Introduce the challenge of working backwards from max/min points to find the equation.
  • Graphing Max and Min (5 mins)
    Explain the process of plotting the given maximum and minimum points on a coordinate plane. Emphasize that the distance between these points represents half the period of the sinusoidal function.
  • Calculating the Period and B Value (10 mins)
    Explain the formula: Period = 2π/B. Demonstrate how to calculate the period from the max/min points and then solve for the B value. Provide example calculations.
  • Finding the Midline and K Value (10 mins)
    Explain the concept of the midline and how it relates to the vertical shift (K value). Show the formula: Midline = (Max + Min) / 2. Calculate the midline and K value using the given max/min points. Emphasize its representation on the graph.
  • Determining the Amplitude (5 mins)
    Define amplitude as the distance from the midline to the maximum or minimum point. Show the formula: Amplitude = |(Max - Min) / 2|. Calculate the amplitude using the given max/min points.
  • Finding the Phase Shift (H Value) for Cosine (10 mins)
    Explain that cosine functions typically start at a maximum. Determine the horizontal shift (H value) needed to align the standard cosine function with the given graph. Note that a shift left corresponds to a positive H value in the equation and vice versa.
  • Writing the Cosine Equation (5 mins)
    Combine the calculated A, B, H, and K values to write the complete cosine equation: y = A cos(B(x - H)) + K. Show the final equation.
  • Relating Sine and Cosine: Finding the Phase Shift (H Value) for Sine (10 mins)
    Explain the relationship between sine and cosine as horizontal shifts of each other. Determine the phase shift needed to transform the cosine graph into a sine graph. Explain that the phase difference is a quarter of the period. Calculate the new H value for the sine equation.
  • Writing the Sine Equation (5 mins)
    Combine the calculated A, B, H, and K values to write the complete sine equation: y = A sin(B(x - H)) + K. Show the final equation.

Interactive Exercises

  • Graph Matching
    Provide students with several sinusoidal equations and corresponding graphs (without labels). Have them match the equations to the correct graphs, justifying their choices based on amplitude, period, phase shift, and midline.
  • Equation Derivation Practice
    Give students sets of maximum and minimum points and have them work in pairs to derive both sine and cosine equations that model the data. Encourage them to discuss their reasoning and compare their solutions.

Discussion Questions

  • How does changing the amplitude affect the graph of a sinusoidal function?
  • What is the relationship between the period of a sinusoidal function and its B value?
  • How can you determine whether a sinusoidal function should be modeled using sine or cosine?
  • Explain the impact of the phase shift on the graph of a sinusoidal function.
  • Can you find multiple sine equations for the same graph? Explain.

Skills Developed

  • Analytical Thinking
  • Problem Solving
  • Graphing and Visualization
  • Mathematical Modeling

Multiple Choice Questions

Question 1:

Given a sinusoidal function with a maximum at (0, 5) and a minimum at (2, 1), what is the amplitude?

Correct Answer: 2

Question 2:

For the same function, what is the midline (vertical shift)?

Correct Answer: y = 3

Question 3:

What is the period of the sinusoidal function described above?

Correct Answer: 8

Question 4:

Which of the following transformations is necessary to convert a cosine function to a sine function?

Correct Answer: Phase shift

Question 5:

The general form of a sinusoidal equation is y = A sin(B(x - H)) + K. What does 'B' represent?

Correct Answer: Period

Question 6:

If the period of a sinusoidal function is 6π, what is the value of 'B' in the equation?

Correct Answer: 1/3

Question 7:

A sinusoidal function has a maximum at (π/2, 7) and a minimum at (3π/2, 3). What is the vertical shift?

Correct Answer: y = 5

Question 8:

Given the sinusoidal equation y = 4 cos(2x - π) + 1, what is the phase shift?

Correct Answer: π/2

Question 9:

How does a negative amplitude affect the graph of a sinusoidal function?

Correct Answer: Reflection over the x-axis

Question 10:

Which parameter affects the horizontal positioning of a sinusoidal function?

Correct Answer: Phase Shift

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 horizontal shift of a sinusoidal function is known as the __________ __________.

Correct Answer: phase shift

Question 3:

The formula to find the midline is (Max + Min) / __________.

Correct Answer: 2

Question 4:

The relationship between the period and the B value is given by the formula: Period = 2π / __________.

Correct Answer: B

Question 5:

If a cosine function starts at its maximum value, then the phase shift is __________.

Correct Answer: 0

Question 6:

Sine and cosine functions are out of phase by a __________ of a period.

Correct Answer: quarter

Question 7:

A negative amplitude causes a __________ over the x-axis.

Correct Answer: reflection

Question 8:

The vertical shift of a sinusoidal function is also called the __________.

Correct Answer: midline

Question 9:

The B value in a sinusoidal equation affects the __________ of the function.

Correct Answer: period

Question 10:

In the equation y = A sin(B(x - H)) + K, the 'K' value represents the __________ __________.

Correct Answer: vertical shift

Educational Standards

PC.T.2.1:Create models for situations involving trigonometry. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.T.1.8:Graph all six trigonometric functions, identifying key features.

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