C HAPTER 7: T RIGONOMETRIC G RAPHS 7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Essential Question: What translation is related to a phase shift in a trigonometric graph?

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Vertical Changes As we talked about previously, vertical changes occur either before or after the function (away from the x) The graph of k ( t ) = -2 cos t + 3 represents: A vertical reflection (the “-” in “-2”) A vertical stretch by a factor of 2 (vertical acts as expected) A vertical shift up 3 units

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Phase Shifts Phase shifts are simply horizontal shifts, with one exception: Phase shifts don’t have a direction (left/right). Shifts to the left are negative phase shifts Shifts to the right are positive phase shifts Example: The graph of g ( t ) = sin( t + π / 2 ) represents a shift to the left π / 2 units (horizontals work opposite as expected), so we say it has a phase shift of - π / 2. Example #2: What is the phase shift for the graph of h ( t ) = cos( t – 2π / 3 ) 2π / 3

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Combined Transformations State the amplitude, period, and phase shift of f ( t ) = 3 sin (2 t + 5) Before we can identify horizontal changes, remember that the inside parenthesis must begin with ONLY “t” So divide all terms inside the parenthesis by 2, and push that number outside as a GCF f ( t ) = 3 sin 2( t + 5 / 2 ). Now we can identify our transformations. Amplitude: Period: Phase Shift: 3 2π ½ = π -5 / 2

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Combined Transformations State the amplitude, period, vertical shift and phase shift of g ( t ) = 2 cos (3 t – 4) – 1 Pull out GCF: Amplitude: Period: Vertical Shift: Phase Shift: 2 2π 1 / 3 = 2 π / 3 4 / 3 g ( t ) = 2 cos 3( t – 4 / 3 ) – 1 -1

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Identifying Graphs (sin) Find a sine function and a cosine function whose graphs look like the graph below Because the gap from maximum to minimum is 4 (2 to -2), the graph has an amplitude of 4/2 = 2 For a sin graph, the function would start at 0. That is π / 4 away, so it has a phase shift of π / 4. f ( t ) = 2 sin (t – π / 4 )

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Identifying Graphs (cos) Find a sine function and a cosine function whose graphs look like the graph below The amplitude is still 2 For a cos graph, the function would start at 1. That is 3π / 4 away, so it has a phase shift of 3π / 4. f ( t ) = 2 cos (t – 3π / 4 )

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Possible Identities Which of the following equations could possibly be an identity? cos( π / 2 + t) = sin t cos( π / 2 – t) = sin t Graph each of the equations. Identities will overlap. Which of the following equations could possibly be an identity? cot t / cos t = sin t Graph as “(1/tan t) / cos t” sin t / tan t = cos t Graph each of the equations. Identities will overlap. → I dentity → I dentity

7.4: P ERIODIC G RAPHS AND P HASE S HIFTS Assignment Page 508 – 509 Problems 1 – 39, 55 – 57 (odd problems)