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Elementary Functions

by: August Feeney

Elementary Functions MTH 112

Marketplace > Portland Community College > Math > MTH 112 > Elementary Functions
August Feeney
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This 4 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 112 at Portland Community College taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/224645/mth-112-portland-community-college in Math at Portland Community College.


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Date Created: 10/19/15
Haberman MTH 112 Section 1 Periodic Functions and Trigonometry qm x nwyov at wanlt vmvo1un uv 39v w w Module 2 Introduction to Periodic Functions DEFINITION A function f is periodic if its values repeat on regular intervals Hence f is periodic if there exists some constant c such that ftcft for all t in the domain of f such that ft c is defined Recall that this means that if the graph of y ft is shifted horizontally c units then it will appear unaffected Any activity that repeats on a regular time interval can be described as periodic For example if the bell at a local church rings once everyhouronthehour then the function that relates the time of day to whether or not the bell will ring is a periodic function Similarly if you take your dogs on a onehour walk every day at 10 am then the function that associates the time of day with whether or not you re on a walk with your dogs is a periodic function EXAMPLE 1 The following are graphs of periodic functions We know that they are periodic since an interval of each graph repeats overandoverandover that interval has been highlighted green in the graphs below A s1vsvm uKN AAAI 3V 1 13V5V rw J that ft c ft for all t in the domain of f such that ft c DEFINITION The period of a periodic function f is the smallest value lcl such is defined EXAMPLE 2 Find the period of the functions graphed below 1 A a L s y7 Vs Wme39 4 The period of this function is 8 units since we can shift the graph horizontally 8 units the graph will appear unaffected Notice that the green intervalquot represents one period and is 8 units long hKNARAAz 2 1v1vsvsvvv The period of this function is 2 units since we can shift the graph horizontally 2 units the graph will appear unaffected Notice that the green intervalquot represents one period and is 2 units long DEFINITIONS I The midline of a periodic function is the horizontal line midway between the function s minimum and maximum values If y ft is periodic and fmax and fmin are the maximum and minimum values of f respectively then the equation of the midline is y The amplitude of a periodic function is the distance between the function s maximum value and the midline or the function s minimum value and the midline 4 EXAMPLE 3 Find the midline and the amplitude of the functions graphed below ER 4zi81215W24 The midline of this function is the t axis ie the line y 0 since the maximum output for the function is 4 while the minimum output is 74 and 424 0 The amplitude of this function is 4 units Jquot 4 A A2 A NA Y 39 quotT I T at 3i 1j13 3V 5 6 TV The midline of this function is the line y 1 since the maximum output for the function is 4 while the 472 TL minimum output is 72 and The amplitude of this function is 3 units EXAMPLE 4 SOLUTION The Amusement Park has a Ferris wheel 200 feet in diameter The wheel rotates at a constant rate and completes a rotation once every 40 minutes Let ht represent the height in feet of a Ferris wheel passenger t minutes after boarding the wheel at ground level Sketch a graph of y W Since the Ferris wheel completes a rotation once every 40 minutes the values of the height function y ht will repeat every 40 minutes so the period of y ht is 40 minutes As the Ferris wheel rotates the passenger will start at ground level and then climb 200 feet to the top of the wheel this is half of a rotation so it will take 20 minutes During the next 20 minutes the passenger will descend from 200 feet to ground level Then during next rotation these values will repeat so that the passenger will be at ground level after 80 minutes and at the height of 200 feet after 60 minutes We can summarize this information in the table below tminutes 0 20406080 htfeet 0 2oo 0 200 0 Let s plot the order pairs t ht on the coordinate plane in Figure 1 below 200 y o o 1 50 100 50 f 20 4390 60 8390 Figure 1 Some points on the graph of y ht Now we need to connect the dots To determine how the dots should be connected let s imagine that we are the Ferris wheel s passengers To help you get a good mental image of the trip around the wheel just imagine traveling around a circle see Figure 2 below Figure 2 Simplified Ferris wheel When we first begin to travel around the wheel starting at ground level ie at the 6 o clock position on the wheel at first we don t gain much elevation After a short period near the tip of the red arrow in Figure 2 we begin to gain elevation more and more quickly Onequarter of the way around the wheel we ll be gaining elevation most quickly since the wheel is vertical here As we near the top of the wheel it gets flatter and flatter so we ll begin to gain less and less elevation until we reach the top of the wheel This tells us that our graph of y ht should be steep between the dots we ve drawn in Figure 1 but get less and less steep as it approaches the dots Let s use this information to connect the dots on our graph 200 y 1 so 100 50 3 20 4 60 8390 Figure 3 The graph of y ht


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