4 8 Translation of Circular Functions fguilbert fguilbert

4 -8 Translation of Circular Functions fguilbert

fguilbert

A phase shift moves a wave to the left or right. It’s like moving the Y axis. 2 2 fguilbert

A phase shift moves a wave to the left or right. y = A sin ( b x - C) + D Phase Shift PS = C b fguilbert

Remember the rules from basic parent graphs. Graph 2 Y=X fguilbert

Remember the rules from basic parent graphs Graph 2 Y = (X+2) fguilbert

What does +2 in ( ) do ? Y= 2 X Y= (X + fguilbert 2 2)

+2 in ( ) moves it left 2 Y= 2 X Y= (X + fguilbert 2 2)

Moves each point 3. What does - 3 in (right ) do ? Y= 3 X Y= (X - 3 fguilbert 3 )

This happens with waves too. y= Cos ( lx - ) y = A cos ( b x- C)+ D PS= C = Find the b 1 Phase shift ? = + fguilbert

y= Cos ( X - ) If the phase shift is , each point on the cosine graph is moved + or moved right units. fguilbert

Graph y = Cos ( X - ) Find Amplitude l A l= 1 Period 2 =2 b Phase shift c = b fguilbert

Graph y = Cos ( X - ) A= 1 P= 2 PS= Use PS to move the Y axis by the amount of the PS. fguilbert

Graph y = Cos ( X - ) A= 1 P= 2 PS= Use A and P to graph the beginning and end of the w 2 fguilbert 3

Graph y = Cos ( X - ) A= 1 P= 2 PS= 1/2 way between graph the minimum. 2 fguilbert 3

Graph y = Cos ( X - ) A= 1 P= 2 PS= Draw a smooth curve 2 fguilbert 3

Graph y = Cos ( X - ) A= 1 P= 2 PS= The graph continues to the left and right. fguilbert

y = A sin ( b x - C) + D D tells us the vertical shift fguilbert

y = A sin ( b x - C) + D Graph y = 2 sin x + 1 not the same as y = 2 sin (x + 1) fguilbert

no vertical shift up 1 y= 2 sin. X y=2 sin. X +1 Like moving the x axis up fguilbert

y = sin x y = sin (X - /2 ) fguilbert

y = sin x y = sin (X - /2 ) 2 /2 = 1. 57 fguilbert

Theorem For all real numbers X cos ( x - /2 ) = sin x fguilbert

Graph these on your calculator. y = cos ( x - /2 ) y = sin x fguilbert

cos ( x - /2 ) = sin x cos ( x - /2 ) sin x fguilbert

Write an equation if the amp = 3, 0 period is 180 , 0 phase shift is 90 , and vertical shift is - 4 fguilbert

Write an equation 0 A = 3, P= 180 , 0 PS =90 , VS = - 4 y =Asin ( b x - C) + D 3 -4 Still need b and C fguilbert

Write an equation A=3, P= 0 0 180 , PS=90 , Find b per = 360 b VS=- 4 180 = 360 b 2=b fguilbert

Write an equation A=3, P= Find 0 0 180 , PS=90 , VS=- 4 C PS = c b 90 = c 2 180 = c fguilbert

Write an equation A=3, P= 0 0 180 , PS=90 , VS=- 4 y =A sin ( b x - C) + D y = 3 sin (2 x -180) -180 - 4 fguilbert

Thought for the day: Don’t make waves or you’ll rock the boat. fguilbert

Electricity & phase shifts. In an AC circuit, there are 2 waves -- voltage & current flow. If the waves coincide, they are IN PHASE. fguilbert

In Phase fguilbert

If current lags behind voltage, then INDUCTANCE is created. 2 fguilbert

Voltage is black Current is red Current lags behind. Look at the y values. 2 fguilbert

Max inductance occurs when the current lags behind the voltage by /2. Assume both waves have the same Amp & Period, fguilbert

and voltage is modeled by y = sin x. In a situation of Max Inductance, find an equation for the current and sketch the 2 waves. fguilbert

In max inductance, Current has a phase shift of /2. y = sin (X - /2 ) This is not efficient. It’s better to have them in phase. fguilbert