Exploring Periodic Data. Objectives To identify cycles and periods of periodic functions To find the amplitude of periodic functions

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CC-3 Eploring Periodic Data Common Core State Standards MACC.9.F-IF.

do the situations shown have in common? Eplain. Another eample: 7-ear locusts!

ne complete pattern is a ccle. A ccle ma begin at an point on the graph of the function.

The -value in a periodic function often represents time.

intervals of constant length. Problem Identifing Ccles and Periods Analze the periodic function below.

1 CC-3 Eploring Periodic Data Common Core State Standards MACC.9.F-IF.. For a function that models a relationship between two quantities, interpret ke features of graphs... and sketch graphs... Also Prepares for MACC.9.F-TF..5 MP, MP, MP 3, MP, MP bjectives To identif ccles and periods of periodic functions To find the amplitude of periodic functions What do the situations shown have in common? Eplain. Another eample: 7-ear locusts! MATHEMATICAL PRACTICES A periodic function is a function that repeats a pattern of -values (outputs) at regular intervals. ne complete pattern is a ccle. A ccle ma begin at an point on the graph of the function. The period of a function is the horizontal length the distance along the -ais of one ccle. The -value in a periodic function often represents time. Lesson Vocabular periodic function ccle period amplitude Essential Understanding Periodic behavior is behavior that repeats over intervals of constant length. Problem Identifing Ccles and Periods Analze the periodic function below. Identif the ccle in two different was. What is the period of the function? Is there a good point at which to start the ccle? if ou start at the maimum value, it is eas to tell when ou have completed the ccle. Begin at an point on the graph. Trace one complete ccle. one ccle one ccle 5 period 9 3 period hsmase_00_a9 The beginning and ending -values of each ccle determine the period of the function. Ted Smkal 0/05/09 ne ccle begins at = and ends at = ; - =, so the period of the 0/0/09 function is. 9 9 Chapter 3 Periodic Functions and Trigonometr HSM5_AHon_SE_CC_3_TrKit.indd 9 Common Core HSM5_A

2 Got It?. Analze each periodic function. Identif the ccle in two different was. What is the period of the function? a. b. 3 3 You can analze the graph of a function to determine if the function is periodic. Problem Identifing Periodic Functions Do the -values of the function repeat? No; there is a repeating pattern but the actual -values do not repeat. Hon_SE_CC_3_TrKit.indd 99 /5/3 Is the function periodic? If it is, what is its period? A Although the graph shows similar curves, the -values from one section do not repeat in other sections. The function is not periodic. B 0 The pattern of -values in one section repeats eactl in other sections. The function is periodic. (0, ) (0, ) 0 Find points at the beginning and end of one ccle. Subtract the -values of the points: 0-0 = 0. The pattern in the graph repeats ever 0 units, so its period is 0. Got It?. Is the function periodic? If it is, what is its period? a. b. c. Reasoning If the period of a function is seconds, how man ccles does it have in a minute? What is the period of a function that has 0 ccles per minute (for eample, a point on a spinning wheel)? That has 0 ccles per second (for eample, a point on the end of a tuning fork)? Lesson 3- CC-3 Eploring Periodic Data Eploring Periodic Data 99 /5/3 99

3 The amplitude of a periodic function measures the amount of variation in the function values. maimum amplitude amplitude midline minimum The midline is the horizontal line midwa between the maimum and minimum values of a periodic function. The amplitude is half the difference between the maimum and minimum values of the function. amplitude 5 (maimum value minimum value) Problem 3 Finding Amplitude and Midline of a Periodic Function What is the amplitude of the periodic function at the right? What is the equation of the midline? maimum 0 minimum Use the definition of amplitude. amplitude = (maimum value minimum value) Substitute for the maimum and - for the minimum. = [ ( )] Subtract within parentheses and simplif. = () = 3 The midline is the horizontal line through the average of the maimum and minimum values. = (maimum value + minimum value) = [ + ( )] Substitute the values and solve. = () = Got It? 3. What is the amplitude of each periodic function? What is the equation of the midline? a b. 5 Chapter 3 Periodic Functions and Trigonometr HSM5_AHon_SE_CC_3_TrKit.indd 00 Common Core HSM5_A

4 You can model some data with periodic functions. The rotation of a Ferris wheel, the beating of a heart, and the movement of sound waves are all eamples of real-world events that generate periodic data. Problem Using a Periodic Function to Solve a Problem Sound Waves Sound is produced b periodic changes in air pressure called sound waves. The ellow graph in the digital wave displa at the right shows the graph of a pure tone from a tuning fork. What are the period and the amplitude of the sound wave? How does identifing the ccle help ou? The period is the horizontal length of the ccle. The amplitude is half the vertical length of the ccle. ne ccle of the sound wave occurs from 0.00 s to 0.00 s. The maimum value of the function is.5, and the minimum value is.5. Find the period. period = = 0.00 STEM Find the amplitude. amplitude = (.5 -.5) = () = The period of the sound wave is 0.00 s. The amplitude is. Got It?. What are the period, the amplitude, and the equation of the midline of the green graph in the digital wave displa in Problem? Lesson Check Do ou know HW? Do ou UNDERSTAND? 3. Writing A sound wave can be graphed as a periodic function. Name two more real-world eamples of periodic functions. Determine if the function is or is not periodic. If it is, find the period.. 3 MATHEMATICAL PRACTICES. Error Analsis A student looked at the following function and wrote that the amplitude was. Describe and correct the student s error. 0. Hon_SE_CC_3_TrKit.indd 0 /5/3 π π 3π 5. Reasoning Suppose f is a periodic function. The period of f is 5 and f () =. What are f () and f ()? Eplain our reasoning.. A wave has a maimum of. If its midline is at =, what is its minimum? Lesson 3- CC-3 Eploring Periodic Data Eploring Periodic Data 0 /5/3 0

5 MATHEMATICAL Practice and Problem-Solving Eercises A Practice PRACTICES See Problem. Identif one ccle in two different was. Then determine the period of the function See Problem. Determine whether each function is or is not periodic. If it is, find the period Appl See Problems 3 and. Find the amplitude of each periodic function, and midline... B. Sketch the graph of a sound wave with the given period, amplitude, and midline. 9. period 0.0, amplitude, midline 0. period 0.005, amplitude 9, midline 5. Complete each statement with or. a. You use -values to compute the amplitude of a function. b. You use -values to compute the period of a function.. Which of the following could be represented b a periodic function? Eplain. a. the average monthl temperature in our communit, recorded ever month for three ears b. the population in our communit, recorded ever ear for the last 50 ears c. the number of cars per hour that pass through an intersection near where ou live, recorded for two consecutive work das 0 0 Chapter 3 Periodic Functions and Trigonometr HSM5_AHon_SE_CC_3_TrKit.indd Common0Core HSM5_A /5/

6 3. Writing What do all periodic functions have in common?. Think About a Plan A person s pulse rate is the number of times his or her heart beats in one minute. Each ccle in the graph represents one heartbeat. What is the pulse rate? Will ou compute the period or the amplitude, or both? Does the graph provide information ou do NT need? RHYTHM STRIP unit (horizontal) = 0. s unit (vertical) = 0.5 mv C Challenge 5. Health An electrocardiogram (EKG or ECG) measures the electrical activit of a person s heart in millivolts over time. Refer to the graph in the previous eercise. a. What is the period of the EKG shown above? b. What is the amplitude of the EKG?. pen-ended Sketch a graph of a periodic function that has a period of 3 and an amplitude of. Find the maimum, minimum, and period of each periodic function. Then cop the graph and sketch two more ccles Language Arts Functions that repeat over time are common in everda life. The English language has man words that stand for common periods of time. State the period of time from which each term derives. 30. annual 3. biweekl 3. quarterl 33. hourl 3. circadian 35. Suppose g is a periodic function. The period of g is, g (3) = 7, and g () = 70. Find each function value. a. g (7) b. g (0) c. g (-) d. g (5) 3. Calendar A da is a basic measure of time. A solar ear is about 35. das. We tr to keep our calendar in step with the solar ear. a. If ever calendar ear has 35 das, b how man das would the calendar ear and the solar ear differ after 00 ears? b. If ever fourth ear has an etra leap da added, b how man das would the two sstems differ after 00 ears? c. If ever hundred ears the leap da is omitted, b how man das would the two sstems differ after 00 ears? d. Reasoning Wh is it important for the difference between the calendar ear and the solar ear to be zero? CC-3 Eploring Periodic Data 03

Chapter 14 Trig Graphs and Reciprocal Functions Algebra II Common Core

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