# 3.1: Getting’ the Swing

**At Grade**Created by: CK-12

*This activity is intended to supplement Calculus, Chapter 2, Lesson 4.*

## Part 1 – Warm-up

In \begin{align*}y1\end{align*}, enter \begin{align*}\cos(x)\end{align*}. Press \begin{align*}F3\end{align*} and select **7: ZoomTrig**. Use the graph to answer the following questions.

1. What is the range?

2. What is the amplitude? \begin{align*}A=\end{align*}

3. What is the period? \begin{align*}T=\end{align*}

Now change your calculator mode to split screen. Press 3 and select **TOP-BOTTOM** for **Split Screen**. For **Split 1 App**, select \begin{align*}Y=\end{align*} **Editor**. For **Split 2 App**, select **Graph**. In \begin{align*}y2\end{align*}, enter an equation in the form \begin{align*}y = A \cdot \cos(B \cdot x) + C\end{align*}, where \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} are integers. Press \begin{align*}2 + \alpha\end{align*} to swap applications to see the graph screen update. Press \begin{align*}2 + \alpha\end{align*} again to go back to the \begin{align*}Y=\end{align*} **Editor** to modify your equation. To answer the following questions, modify the corresponding variable to observe the changes each variable has to the equation.

4. Describe the effect of increasing \begin{align*}A\end{align*}.

5. Describe the effect of increasing \begin{align*}C\end{align*}.

6. Describe the effect of increasing \begin{align*}B\end{align*}.

7. What is the relationship between \begin{align*}B\end{align*} and the period, \begin{align*}T\end{align*}?

8. If a positive \begin{align*}D\end{align*} shifts the graph to the right \begin{align*}D\end{align*} units, what is the general sinusoidal equation for which this is true?

## Part 2 – Collect & Analyze Data

You will collect data of a pendulum swinging. Using the skills reviewed in the warm-up, write a cosine function that models the data collected. Estimate the amplitude and period and phase shift, \begin{align*}D\end{align*}, to the nearest tenth. If a motion detector is not available, use the lists *time, distance*, and *velocity* from your teacher and graph a function to model that data. To collect data, complete the following steps:

- Using an I/O cable, connect the motion detector to the graphing calculator.
- On the
**HOME**screen, run the*Ranger*program. Select**1:Setup/Sample….**Use the settings that appear to the right and press**ENTER**. - Position the motion detector so that it is facing the pendulum, swing the pendulum, and press
**ENTER**to begin collecting data.

- If your data doesn’t look sinusoidal, press
**ENTER**and select**3:Repeat Sample**to repeat the trial. Then press**ENTER**to begin collecting data again. - Model the distance-time data with a function. Derive the velocity and acceleration equations. Select
**7:Quit**when you are finished.

Record your position, velocity and acceleration equations for your experiment data here:

\begin{align*}y=\\ \\ \\ v=\\ \\ \\ a=\end{align*}

Confirm your position and velocity equations by graphing them. To confirm your position equation, enter your equation in \begin{align*}y1\end{align*} and select to show Plot 1 as shown to the right. To plot the velocity-time graph, use \begin{align*}L1\end{align*} for time and \begin{align*}L3\end{align*} for velocity. For the acceleration-time graph, use \begin{align*}L1\end{align*} for time and \begin{align*}L4\end{align*} for acceleration.

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