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Start Up Day 26 1. Graph each function from -2π to 2π g(x) = cos ( x ) The range of y = sin x is _____________. The range of the cosine curve is ________________. 2. Find a polynomial function with zeros of: 0, -4, 3i 1 by C.Kennedy 5/24/2017 f (x) = sin ( x ) OBJECTIVE: SWBAT use trigonometric graphs to define and interpret features such as domain, range, intercepts, periods, amplitude, phase shifts, vertical shifts and asymptotes. SWBAT to graph Sine and Cosine functions with and without graphing technology. EQ: What are the key points for the basic Sine and Cosine parent graphs & where do they come from? How do the values of “a” ,”b”, “h” and “k” affect the graphs of Sine and Cosine? HOME LEARNING: Worksheet#1 Graphing Sine Functions Graphing Trigonometric Functions Sine Waves: The Movers and the Shapers 3 by C.Kennedy 5/24/2017 The UNIT CIRCLE—unwrapped! • Y=sin x is the basic curve: one hill, one valley, starting at 0 and ending at 2 π • 5 KEY Points: 0, 1, 0, -1 , 0 • Y=cos x is the basic curve: an upside down bell shape, starting at 0 and ending at 2 π • 5 KEY Points: 1, 0, -1, 0 ,1 4 by C.Kennedy 5/24/2017 “h” AND “k” OR “c” & “d”, respectively: The “Movers” • The “h”or the “c” is the most difficult to see! (although it is always in the parentheses!) – The “h” causes a “phase shift” OR “HORIZONTAL TRANSLATION” – You have to factor out your “b” in order to see your “h” for the HT! • The “k”or the “d” is much more obvious! – When looking to the “k”, you get exactly what you see! – The “k” causes a VERTICAL TRANSLATION 5 by C.Kennedy 5/24/2017 y sin x 1 2 a= Amplitude: b= Period/Wavelength: h= Start: k= End: 6 by C.Kennedy 5/24/2017 A and B: The Shapers! • The “b” value: FREQUENCY = b – Horizontal Compression—If the “b” is greater than 1 – Horizontal Stretch –If the “b” is less than one. – Period OR Wavelength = 2pi/b • The “a” value: AMPLITUDE = Absolute Value of a – Vertical stretch--If the absolute value of “a” is greater than one. – Vertical Compression--If the absolute value of “a” is less than one. – Reflection over the x-axis—If the “a” is a negative value. 7 by C.Kennedy 5/24/2017 8 by C.Kennedy 5/24/2017 Where to begin? y=a sin b(x-h)+k 1. First-- think of the basic wave and the 5 KEY POINTS Sine: (0,1, 0,-1, 0) OR Cosine:(1, 0, -1, 0, 1) 2. Next--Identify the values of a,b,”h”(c) and “k” (d) 3. Finally--determine amp, frequency, period or wave length, horizontal and/or vertical translations 9 B y C.Kennedy 5/24/2017 Sketching y=a sin b(x-h)+k 1) Dash in a horizontal axis at “k” or“d” 2) If c=o then start at o and end at 2π/b, otherwise continue with begin/end – BEGIN: Set your (bx-bh) =o and solve for x— this will be your starting point. – END: Set your (bx-bh) =2π and solve for x again—this will be your ending spot. 3) Divide your “wavelength” into FOUR equal spaces— making room for your FIVE KEY PLACES! 4) Let your “a” wipe the “ones” away and sketch your wave! – Remember that a “-a” causes a reflection over the x-axis 10 by C.Kennedy 5/24/2017 y 2 sin 2 x 1 A=-2 START: 2 3p 2x - p = 2p ;x = 2 END: B=2 2x - p = 0;x = p Period/Wavelength: C=-π/2 Amplitude: 2p 2p = =p b 2 a = -2 = 2 D=1 11 by C.Kennedy 5/24/2017 y 2 sin 2 x 1 12 by C.Kennedy 5/24/2017 Let’s apply it—Now you try it! Sketch each over 2 complete periods. () #3. y = -4sin x æ 2x ö #16. y = -4sin ç ÷ è 3ø æxö # 27. y = 4sin ç ÷ è4ø 5/24/2017 by C.Kennedy 13 Start Up Day 27 Problems #58 & 61 from p.358 Construct a sine function (sinusoid) with the given constraints: #58 Amplitude 2, period 3π, point (0, 0) “a” and “b”: The Shapers! Y = a cos b(x – c) +d “a” Amplitude = IaI “b” Frequency = b Period (wavelength)= 2π/b ½ the distance from the Max to the min. a Reflection over x-axis 5/24/2017 IaI> 1, Vertical Stretch IaI < 1, Vertical Compression IbI>1 Horizontal Compression by C.Kennedy IbI<1 Horizontal Stretch 15 “h” and “k” OR “c” & “d” respectively— The Movers y=a cos b(x-h)+k The Movers ”h” Horizontal Translation= “h” or “c” Always opposite of what you see in the ( ) and don’t Forget to factor out your “b” in order to see your REAL “h”OR set your () = 0 and Solve for “x”. Changes the starting point of the 16 by C.Kennedy curve! “k” or “d” K = Vertical Translation k lifts or lowers the base lin of the curve 5/24/2017