Unit #6: Graphs and Inverses of Trig Functions Accel Precalc Unit #6: Graphs and Inverses of Trig Functions Lesson 9: Modeling Real World Data & Simple Harmonic Motion EQ 1: How can you create trig models to represent real world data?

Modeling Data Using Trig Functions:

Objective: Create a model in the form Define the variables in the model: A = B = t = h = k = amplitude new period time phase shift vertical shift

WHY?? What VIEWING WINDOW could you use? The table below lists the normal average high temperatures for Charlotte, NC. Month Temp °F January 1 51.3 July 7 90.1 February 2 55.9 August 8 88.4 March 3 64.1 September 9 82.3 April 4 72.8 October 10 72.6 May 5 79.7 November 11 62.8 June 6 86 December 12 54 Use your graphing calculator to create a scatterplot of the data. What VIEWING WINDOW could you use? WHY??

Unlike the Sinusoids, the horizontal axis of symmetry IS NOT the x-axis!! 2. a) What is the period of the function? _________ 12 months b) Find B (in  radians) . ________ 3. a) What are the minimum and maximum temperatures? Min = _______ Max = _______ 51.3 90.1 19.4 Use these values to find the amplitude. A = _______ How can you use the amplitude and the max or min to find the horizontal axis of symmetry? k = _______ 70.7

Place the value for k in Y1 and graph the horizontal axis of symmetry through your data set. Trace along this line to estimate the closest zero. This value represents the horizontal shift. What is an estimated value for h for this periodic function? h = ____________

5. Put your values for A, B, h, and k into the model equation 5. Put your values for A, B, h, and k into the model equation. Graph this with your data by placing your model equation in Y2. How well does it seem to fit? Model for Normal High Temperature in Charlotte, NC: y = ________________________

Practice Problems: Modeling Data Work on Inverse Trig Worksheets

Day 61 Agenda: DG 25--- 25 minutes Complete Lesson U6 L9

Part II: Simple Harmonic Motion

Equilibrium --- object is at rest; zero point Simple Harmonic Motion Models: RECALL: Equilibrium --- object is at rest; zero point |A| = amplitude Maximum displacement from point of equilibrium

To find c, we must use a given point (t, cos(t)) and solve. Ex. 1 A buoy marking a channel in the harbor bobs up and down as the waves move past. Suppose the buoy moves a total of 6 feet from its high point to its low point and returns to its high point every 10 seconds. Assuming that at t = 0 the buoy is at its high point and the middle height of the buoy’s path is d = 0, write an equation to describe its motion. Should the equation be in terms of sine or cosine? Cosine Buoy starts here. A = 3 10 NP = _____ To find c, we must use a given point (t, cos(t)) and solve. Use (0,3). WHY?

Solve for c.

To find c, we must use a given point (t, sin(t)) and solve. Ex. 2 A weight on a spring bounces a maximum of 8 inches above and below its equilibrium (zero) point. The time for one complete cycle is 2 seconds. Write an equation to describe the motion of this weight, assume the weight is at equilibrium when t = 0. Sine 2 NP = _____ 8 t = 0 8 To find c, we must use a given point (t, sin(t)) and solve. Initial point Use (0,0). WHY?

MODELING DATA & HARMONIC MOTION: Assignment: MODELING DATA & HARMONIC MOTION: Practice Problems: Modeling Data Worksheet: Harmonic Motion