4.2 Period, Amplitude & Phase Shift

Amplitude: If a periodic function has a max value M and a min value m, then amplitude is  aka: half the range/ vertical spread A Sinusoidal Function a, b, c, and d all represent values that affect the basic sine curve in different ways *it doesn’t HAVE to be sine…. could be cosine too!

Period: how long it takes to complete 1 cycle Since the period of sine & cosine are both 2π, determines the factor it is dilated horizontally For The zeros of a graph occur when y = 0. To find them, set your equation equal to 0 Local maximums occur when the graph changes from rising to falling Local minimums occur when it changes from falling to rising is amplitudeand is period

b c a d Amp = Per: IL: = = 0 = –2 = 0 2 –2 a = –2  reflect across x-axis Ex 1) Graph 0  1 st tick mark Check Period: a b

Ex 1 cont) Find at least two zeros: Find local max: cosine has max’s at 0, 2π, etc. Set ‘argument’ = to these argument Find local min: cosine has mins at π, 3π, etc. Set ‘argument’ = to these Let’s confirm with our graph!

2 –2 We were wrong about Max & mins… WHY??? a = –2  reflect across x-axis  Changes Max’s to mins & mins to Max’s Ex 1) Graph Max min Zero Fix it in your notes!!!

*opposite of c horizontal shift (left/ right) *d is it vertical shift (up/ down) Phase Shifts & Vertical Shifts

b c a d Per = IL: = = = 4 = –2 Check Period: Check a point: Factor out b 4 –4 2 –2 –6 Ex 2) Graph  1 st tick mark (easier to add to c)

When x = 0, y = –2 then starts to climb, so use Per = a max to a max Max = –.5 min = –3.5 1 –1 Ex 3) Determine equation from the graph. –2 –4 and shift down 2 a b d

Homework #402 Pg 197 #23, 27, 29, 31 – 34, 36 –45, 49 – 54 all When they ask for frequency, Don’t ignore “word problems”! They give you the equation!!! (Super easy!)