Chapter 7: Trigonometric Graphs 7.4: Periodic Graphs and Phase Shifts

1. Chapter 7: Trigonometric Graphs7.4: Periodic Graphs and Phase Shifts Essential Question: What translation is related to a phase shift in a trigonometric graph?

2. 7.4: Periodic Graphs and Phase Shifts • 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

3. 7.4: Periodic Graphs and Phase Shifts • 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

4. 7.4: Periodic Graphs and Phase Shifts • Combined Transformations • State the amplitude, period, and phase shift off(t) = 3 sin (2t + 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

5. 7.4: Periodic Graphs and Phase Shifts • Combined Transformations • State the amplitude, period, vertical shift and phase shift of g(t) = 2 cos (3t – 4) – 1 • Pull out GCF: • Amplitude: • Period: • Vertical Shift: • Phase Shift: g(t) = 2 cos 3(t – 4/3) – 1 2 2π • 1/3 = 2π/3 -1 4/3

6. 7.4: Periodic Graphs and Phase Shifts • 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. 7.4: Periodic Graphs and Phase Shifts • 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)

8. 7.4: Periodic Graphs and Phase Shifts • 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. → Identity → Identity

9. 7.4: Periodic Graphs and Phase Shifts • Assignment • Page 508 – 509 • Problems 1 – 39, 55 – 57 (odd problems)