Graphing Trigonometric Functions: Unveiling Sine and Cosine Equations

Algebra 2 Grades High School 22:24 Video
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Lesson Description

Learn to derive sine and cosine equations from their graphs. This lesson covers amplitude, period, phase shift, and vertical shift with practical examples.

Video Resource

Write the Sine and Cosine Equation Given the Graph

Mario's Math Tutoring

Duration: 22:24
Watch on YouTube

Key Concepts

  • Amplitude
  • Period and Frequency
  • Phase Shift (Horizontal Shift)
  • Vertical Shift
  • Sine and Cosine as Transformations

Learning Objectives

  • Students will be able to identify the amplitude, period, phase shift, and vertical shift from a given sine or cosine graph.
  • Students will be able to write the equation of a sine or cosine function given its graph.
  • Students will be able to convert between sine and cosine equations for the same graph by adjusting the phase shift.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic shapes of sine and cosine functions. Briefly discuss the transformations: amplitude, period, phase shift and vertical shift.
  • Video Viewing and Note-Taking (15 mins)
    Watch "Write the Sine and Cosine Equation Given the Graph" by Mario's Math Tutoring. Students should take notes on the formulas and methods presented for identifying key parameters from graphs.
  • Guided Practice (20 mins)
    Work through the examples from the video step-by-step. Pause after each step to allow students to process and ask questions. Emphasize how to calculate amplitude, period, phase shift, and vertical shift.
  • Independent Practice (15 mins)
    Provide students with additional graphs of sine and cosine functions. Students should independently determine the corresponding equations. Encourage them to check their answers by graphing the equations on a graphing calculator or online tool.
  • Wrap-up and Discussion (5 mins)
    Recap the key steps for writing sine and cosine equations from graphs. Address any remaining questions or areas of confusion.

Interactive Exercises

  • Graphing Calculator Exploration
    Students use graphing calculators to manipulate the parameters (A, B, H, K) in the general sine and cosine equations and observe the effects on the graph. This reinforces the visual connection between equation and graph.
  • Equation Matching Game
    Create a set of cards with graphs on some cards and corresponding equations on others. Students match the graphs to their equations.

Discussion Questions

  • How does the amplitude affect the graph of a sine or cosine function?
  • What is the relationship between the period and the frequency of a trigonometric function?
  • How can you determine whether a graph is better represented by a sine or a cosine function?
  • Why are there multiple possible equations for the same trigonometric graph?

Skills Developed

  • Analytical Skills
  • Problem-Solving Skills
  • Graph Interpretation
  • Equation Formulation

Multiple Choice Questions

Question 1:

What does the 'A' in the equation y = A sin(Bx - H) + K represent?

Correct Answer: Amplitude

Question 2:

If the maximum value of a sinusoidal function is 7 and the minimum value is 1, what is the amplitude?

Correct Answer: 3

Question 3:

The period of a trigonometric function is the length of:

Correct Answer: One complete cycle

Question 4:

Which parameter in the equation y = A cos(Bx - H) + K affects the horizontal shift of the graph?

Correct Answer: H

Question 5:

A vertical shift in a trigonometric function moves the graph:

Correct Answer: Up or Down

Question 6:

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

Correct Answer: 1/2

Question 7:

The midline of a sinusoidal function is determined by:

Correct Answer: Vertical Shift

Question 8:

Which function typically starts at its maximum value?

Correct Answer: Cosine

Question 9:

A phase shift to the right is represented by which of the following in the equation y = A sin(Bx - H) + K?

Correct Answer: Positive H

Question 10:

If a sine graph is reflected over the x-axis, what changes in the equation?

Correct Answer: Amplitude becomes negative

Fill in the Blank Questions

Question 1:

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

Correct Answer: amplitude

Question 2:

The __________ represents the horizontal shift of a trigonometric function.

Correct Answer: phase shift

Question 3:

The value of K in y = A cos(Bx - H) + K represents the __________.

Correct Answer: vertical shift

Question 4:

The period of a trigonometric function can be calculated using the formula Period = 2π / __________.

Correct Answer: B

Question 5:

If a cosine function is shifted π/2 units to the left, the value of H in the equation will be __________.

Correct Answer: -π/2

Question 6:

A sine function that starts at the midline and increases has a phase shift of __________.

Correct Answer: 0

Question 7:

The formula to calculate amplitude using maximum and minimum values is A = |max - min| / __________.

Correct Answer: 2

Question 8:

The __________ function typically starts at its maximum value on the y-axis.

Correct Answer: cosine

Question 9:

A negative amplitude indicates a reflection over the __________.

Correct Answer: x-axis

Question 10:

If the period of a sine wave is π, then the value of B in y = sin(Bx) is __________.

Correct Answer: 2

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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