Lesson 24

Shifting the Path: Transforming Trig Graphs for Robotic Motion

Is there a way to capture and share information about the movement of a robot arm?

Learning Intentions:

Identify and apply horizontal (phase) and vertical shifts to sine and cosine graphs, write a transformed trigonometric function to match a real robot movement, explain the roles of A, B, C, and D in the function y=Asin⁡(Bx+C)+Dy=Asin(Bx+C)+D or y=Acos⁡(Bx+C)+Dy=Acos(Bx+C)+D, and sketch a motion graph for their robot that includes a shift, connecting mathematical transformations to real-world robotic motion planning.

Today's Concept:
Yesterday, we learned the basic rhythm of the robot. But right now, our robot always starts at the center (0) and swings symmetrically around it.

Real robots don't do that. Your user might need the robot to wave back and forth, but centered around 45°, not 0°. Or they might need it to start its motion later, after another robot has moved out of the way. We need to be able to pick up that basic sine wave and move it around—shift it up, down, left, or right—to match exactly what our user needs. Today, you become the choreographer, writing the exact equation that tells the robot where to start, where to center its movement, and how to move.

Finding the Midline

You can find the midline by taking the average of the maximum and minimum values of the function. If the equation is in the form y=Asin(Bx−C)+D or y=Acos(Bx−C)+D, the midline is the line y=D

Finding the Amplitude

The amplitude is the maximum displacement or distance from the midline to either the peak (maximum) or trough (minimum) of the wave. It represents the "height" of the oscillation.

You can find the amplitude in two ways:
- From Max and Min: Take half the difference between the maximum and minimum values of the function:

Amplitude =(Maximum−Minimum)/2

- From the Equation: If the equation is in the form y=Asin(Bx−C)+D or y=Acos(Bx−C)+D, the amplitude is the absolute value of A

Finding the Period

The period is the length of one complete cycle of the trigonometric function, representing the horizontal distance over which the wave pattern repeats itself.

- From the Graph: Measure the distance between two consecutive maximum points, two consecutive minimum points, or the distance it takes for the wave to complete one full pattern.
- From the Equation: If the equation is in the form y=Asin(Bx−C)+D or y=Acos(Bx−C)+D

For sine and cosine functions, the period is calculated as

2π/|B| (the fraction with numerator 2 pi and denominator the absolute value of B)

For tangent and cotangent functions, the period is calculated as

π/|B| (the fraction with numerator pi and denominator the absolute value of B)

Station 1

Explore It – Predict, Observe & Justify—Understanding Shifts

Station 2

Practice It – Mastering C and D—Phase & Vertical Shifts in Action

Station 3

Build It – Graph That Motion Challenge

Homework

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