Understanding Periodic Functions and Trigonometry Concepts
Explore the common elements and principles behind periodic functions, cycles, and trigonometry in this informative content. Discover how to determine cycles, periods, amplitudes, and more in trigonometric functions. Dive into identifying periodic functions and understanding midlines and amplitudes.
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Presentation Transcript
Periodic Functions and Trigonometry Unit Objectives: Determine exact values for trigonometric functions: with and without a calculator Write and graph trigonometric functions Find amplitude, period, maximums, minimums and phase shifts for periodic functions Model problems using trigonometric functions Today s Objective: I can find a cycle, period and amplitude of periodic function.
What do these situations have in common? Explain Periodic Function: A function that repeats a pattern of outputs (y-values) at regular intervals Cycle: One complete pattern Period: Horizontal length of a cycle distance along x-axis
One cycle: or ? = 8 to ? = 12 ? = 2 to ? = 6 Period: 6 2 = 12 8 = 4 One cycle ? = 2 to ? = 2 ? = 3 to ? = 7 Period: 2 ( 2) = 4
Determine whether function is periodic. If so identify one cycle and determine the period. Not Periodic One cycle ? = 2 to ? = 8 Period: 8 ( 2) =10 One cycle ? = 0 to ? = 3 Period: 3 0 = 3 Not Periodic
Maximum Midline ? = 1 Minimum amp. = 3 Midline: Horizontal line midway between maximum and minimum values Half the difference between maximum and minimum ? =1 2(maximum + minimum) Amplitude: amp. =1 2(max. min.)
One cycle: ? = 0.004 to ? = 0.008 What is the period, the amplitude and the equation of the midline for each sound wave displayed below. Period: 0.008 0.004 = 0.004 Midline: ? = 2 1 2(2.5 1.5) = 0.5 Amplitude: One cycle: ? = 0.006 to ? = 0.012 Period: 0.012 0.006 = 0.006 Midline: ? = 0.75 Amplitude: 1 2(( 0.5) ( 1)) = 0.25 Pg. 832 #7-25 odd, 35, 36
The Sine Function ? = sin? Today s Objective: I can graph the sine function.
? ? = sin? x 0 ? 6 ? 2 5? 6 ? y
? = sin? OR ? = sin? Domain: All real numbers Range: 1 y 1 Period: 2 Amplitude: 1
Period of a Sine Curve ?1= sin? ?3= sin.5? ?2= sin4? Calculator [MODE]: Radians [WINDOW] Xmin = 0 Xmax = 2 Xscl = /2 Ymin = 2 Ymax = 2 Yscl = 0.5 b = # of cycles from 0 to 2 Period of function = 2? ? = sin?? ?
Amplitude of a Sine Curve ?1= sin? ?3= 2sin? ?2= 2sin? Calculator [MODE]: Radians [WINDOW] Xmin = 0 Xmax = 2 Xscl = /2 Ymin = 2 Ymax = 2 Yscl = 0.5 ? = amplitude (stretch) a = reflects graph across x-axis ? = ?sin?
Find amplitude and period for each equation. 1.? = 2sin3? Amplitude = Period = 2. ? =1 2sin? 1 2 or 0.5 Amplitude = 4 2? 3 2 2? 0.25 Period = = 8? 3.. Amplitude = 3 4? 3 =2? Period = ? Write an equation for the graph. ? = 3sin3 2?
Sketching a Sine Curve Graph (2 cycles) Amplitude = 2 1. Find amplitude and period. 2. Plot 5 points: Midline points Beginning, End Middle Amplitude points Max Min 2? 2 ? = 2sin2? = ? Period = 3. Sketch curve. 5 points: midline max midline min midline
Sketching a Sine Curve Graph (2 cycles) Amplitude = 1.5 ? = 1.5sin? 1. Find amplitude and period. 2. Plot 5 points 3. Sketch curve. 2? ? 2 2? Period = = 4 Pg. 856 #13-35 odd 5 points: midline max midline min midline
