TR-42: Trig Graph Variations (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial focuses on understanding the parameters A and D in the general trigonometric equation ( y = A \cdot \cos(B\theta + C) + D ). These parameters are essential for graphing variations of the cosine function. By grasping how A affects amplitude and D influences vertical shifts, you will enhance your skills in graphing trigonometric functions, which is crucial for various math courses, including those in the International A level curriculum.

Step 1: Understanding Amplitude with Parameter A

The amplitude of the cosine function indicates how far the graph stretches above and below its midline.

• The parameter A in the equation represents the amplitude.
• To determine the amplitude:
  • If A is positive, the amplitude is simply the value of A.
  • If A is negative, the amplitude is still the absolute value of A, but the graph is reflected over the midline.

Practical Advice

• For example, if ( A = 3 ), the amplitude is 3, so the graph will reach a maximum of 3 and a minimum of -3.
• If ( A = -2 ), the amplitude remains 2, but the graph will be inverted.

Step 2: Exploring Vertical Shift with Parameter D

The vertical shift determines the midline of the graph and where it oscillates above or below the x-axis.

• The parameter D in the equation represents the vertical shift.
• To find the midline of the graph:
  • Add D to the amplitude and subtract D from it to find the maximum and minimum values of the function.

Practical Advice

• If ( D = 1 ) and ( A = 2 ):
  • Maximum value = ( D + A = 1 + 2 = 3 )
  • Minimum value = ( D - A = 1 - 2 = -1 )
• Graphing will show the function oscillating between 3 and -1.

Step 3: Analyzing the Complete Function

Combining A and D, you can sketch the complete graph of the function.

• Identify the amplitude (A) and the vertical shift (D).
• Determine the maximum and minimum values based on the calculations from Steps 1 and 2.
• Graph the cosine function by marking the midline (y = D), then plotting the maximum and minimum points.

Practical Tips

• Use graph paper for accuracy.
• Mark key points such as maximum, minimum, and intercepts to guide your drawing.

Conclusion

Understanding the parameters A and D in the equation ( y = A \cdot \cos(B\theta + C) + D ) is crucial for accurately graphing trigonometric functions. Remember that A affects the amplitude and reflection of the graph, while D sets the vertical shift, determining the midline. Practice these concepts by graphing different variations of the cosine function to solidify your understanding. As a next step, consider exploring the effects of parameters B and C on the graph for a more comprehensive grasp of trigonometric variations.