Mastering Sinusoidal Equations: From Maxima and Minima to Sine and Cosine

PreAlgebra Grades High School 10:12 Video
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Lesson Description

Learn to write sinusoidal equations (sine and cosine) by analyzing maximum and minimum points. This lesson covers amplitude, vertical shift, period, and phase shift, enabling you to express equations in both sine and cosine forms.

Video Resource

Write a Sine or Cosine (Sinusoidal) Equation Given the Maximum and Minimum

Mario's Math Tutoring

Duration: 10:12
Watch on YouTube

Key Concepts

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

Learning Objectives

  • Calculate the amplitude, period, vertical shift, and phase shift of a sinusoidal function given its maximum and minimum points.
  • Write the equation of a sinusoidal function in both sine and cosine forms based on its characteristics.
  • Visually represent sinusoidal functions and relate key features to their corresponding equation parameters.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general forms of sine and cosine functions: y = A sin(B(x - H)) + K and y = A cos(B(x - H)) + K, where A is the amplitude, B relates to the period, H is the horizontal shift, and K is the vertical shift. Briefly discuss the significance of maximum and minimum points on a sinusoidal graph.
  • Finding Period from Max/Min (10 mins)
    Explain that the distance between the maximum and minimum represents half the period. Demonstrate how to calculate the period by subtracting the x-coordinates of the max and min and then doubling the result. Use the formula Period = 2π/B to find the value of B.
  • Calculating Amplitude and Vertical Shift (10 mins)
    Introduce the formulas for amplitude: A = (Max - Min)/2, and vertical shift: K = (Max + Min)/2. Work through an example, showing how to apply these formulas to find A and K.
  • Determining Phase Shift (10 mins)
    Explain how the phase shift (H) differs for sine and cosine functions. For cosine, use the x-coordinate of the maximum point. For sine, visualize the sine wave and determine the horizontal shift from the midline. Discuss how shifting to the left involves addition and shifting to the right involves subtraction within the function argument (x-H).
  • Writing the Equations (10 mins)
    Combine all the calculated values (A, B, H, K) to write the sinusoidal equations in both sine and cosine forms. Emphasize that multiple correct answers are possible depending on the chosen starting point on the graph.
  • Practice Problems (10 mins)
    Provide students with additional max/min points and have them determine the corresponding sinusoidal equations in both sine and cosine forms. Encourage them to sketch the graph to aid their understanding.
  • Video Reinforcement (5 mins)
    Watch the assigned video, taking note of Mario's methodology. Then, compare it to the methods discussed during the lesson.

Interactive Exercises

  • Graphing Challenge
    Provide students with sinusoidal equations and have them identify the max/min points and sketch the graph. Then, give the students the graph with the max/min points and have the students create the equation for the graph.

Discussion Questions

  • How does changing the amplitude affect the graph of a sinusoidal function?
  • Explain the relationship between the period and the B value in the sinusoidal equation.
  • Why are there multiple possible equations for the same sinusoidal graph?
  • How does the video's approach compare to ours?

Skills Developed

  • Analytical Thinking
  • Problem-Solving
  • Visual Representation
  • Mathematical Modeling

Multiple Choice Questions

Question 1:

What does the amplitude of a sinusoidal function represent?

Correct Answer: The distance from the midline to the maximum (or minimum) point.

Question 2:

The period of a sinusoidal function is the distance:

Correct Answer: It takes for the function to complete one full cycle.

Question 3:

How is the vertical shift (K) calculated given the maximum and minimum values?

Correct Answer: K = (Max + Min) / 2

Question 4:

What is the formula to determine the 'B' value when given the period?

Correct Answer: B = 2π / Period

Question 5:

A sinusoidal function has a maximum at (π/2, 5) and a minimum at (3π/2, 1). What is the amplitude of this function?

Correct Answer: 3

Question 6:

For a cosine function, where is a good starting point for graphing when determining the phase shift?

Correct Answer: The maximum point

Question 7:

What effect does a '+ H' inside the argument of a sine function have on the graph?

Correct Answer: Shifts the graph to the left by H units

Question 8:

Given a maximum of 7 and a minimum of -1, what is the midline (vertical shift)?

Correct Answer: 3

Question 9:

What does a negative amplitude indicate?

Correct Answer: A reflection over the x-axis

Question 10:

If the distance between a maximum and a minimum on a sinusoidal graph is π, what is the period of the function?

Correct Answer: 2π

Fill in the Blank Questions

Question 1:

The distance between the maximum and minimum of a sinusoidal function is equal to _______ of the period.

Correct Answer: half

Question 2:

The formula for calculating the amplitude (A) is A = (Max - Min) / _______.

Correct Answer: 2

Question 3:

The 'B' value in the equation y = A sin(Bx) is used to calculate the ________ of the function.

Correct Answer: period

Question 4:

A positive 'H' value in the equation y = A cos(x - H) indicates a phase shift to the ________.

Correct Answer: right

Question 5:

If the period of a function is 4π, then the B value is _______.

Correct Answer: 1/2

Question 6:

The _______ shift determines the midline of a sinusoidal function.

Correct Answer: vertical

Question 7:

The maximum point of a cosine function corresponds to a _______ on the graph.

Correct Answer: peak

Question 8:

The general form of a sinusoidal equation can be expressed as y = A * sin(B(x - H)) + ______.

Correct Answer: K

Question 9:

The _______ shift moves a graph horizontally.

Correct Answer: phase

Question 10:

When writing a sine or cosine function, if the graph starts at a minimum point, the _______ may be negative.

Correct Answer: amplitude

Educational Standards

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.2.1:[Objective] Create models for situations involving trigonometry. PC.T.1.8:[Objective] Graph all six trigonometric functions, identifying key features.

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