The Sine Graph: Introduction and Transformations 26 April 2011

The Sine Graph – A Review sin(t) = ytsin(t)

Key Features of y = sin(t) Maximum: Minimum: Domain: Range: 2 – 2 0

Multiple Revolutions tsin(t) 0 0.5π 1π 1.5π 2π 2.5π 3π 3.5π 4π

Trigonometric Graphs Repeat!!! Range: Domain:

Periodicity Trigonometric graphs are periodic because the pattern of the graph repeats itself How long it takes the graph to complete one full wave or revolution is called the period 0 2 –2 1 Period Period: π

Periodicity, cont.

Your Turn: Complete problems 1 – 3 on the Identifying Key Features of Sine Graphs Handout

Calculating Periodicity If f(t) = sin(bt), then period = Period is always positive 1. f(t) = sin(–6t)2. 3.

Your Turn: Calculate the period of the following graphs: 1. f(t) = sin(3t)2. f(t) = sin(–4t) 3.4. f(t) = 4sin(2t) 5.6.

Amplitude Amplitude is a trigonometric graph’s greatest distance from the x-axis. Amplitude is always positive.  If f(t) = a sin(t), then amplitude = | a |

Calculating Amplitude Examples 1. f(t) = 6sin(4t)2. f(t) = –5sin(6t) 3.4.

Your Turn: Complete problems 4 – 9 on the Identifying the Key Features of Sine Graphs handout

Sketching Sine Graphs – Single Smooth Line!!!

Transformations Investigation – Investigation #1 tf(t) = sin(t)f(t) = sin(t) + 3 0

Refection Questions 3.What transformations did you see? 4. A. B. 5.A. B.

Transformations f(t) = a sin(bt – c) + k Vertical Shift Pay attention to the parentheses!!!

Investigation #2! tf(t) = sin(t)f(t) = 2sin(t) 0

Reflection Questions 4. What transformation did you see? Stretch = coefficient is a whole # Compression = coefficient is a fraction 5.A. B. C. 6.A. B. C.

Transformations f(t) = a sin(bt – c) + k Vertical Shift Stretch or Compression “Amplitude Shift” Pay attention to the parentheses!!!

Reflection Questions 4. What transformation did you see? 5.A. B. C. 6.A. B. C.

Transformations f(t) = a sin(bt – c) + k Vertical Shift Stretch or Compression “Amplitude Shift” Pay attention to the parentheses!!! Period Shift

Reflection Questions 4. What transformation did you see? 4.A. B. C. 6.A. B. C.

Transformations f(t) = a sin(bt – c) + k Vertical Shift Stretch or Compression “Amplitude Shift” Pay attention to the parentheses!!! Period Shift Phase Shift

Identifying Transformations f(t) = 2 sin(4t – π) – 3 “Amplitude Shift”: Period Shift: Phase Shift: Vertical Shift: “Amplitude Shift”: Period Shift: Phase Shift: Vertical Shift:

Your Turn: Identify the transformations of the following sine graphs: 1. f(t) = 3 sin(t) f(t) = –sin(t – 4)

Sketching Transformations Step 1: Identify the correct order of operations for the function 1. Period Shifts 2. Phase Shifts 3. Trig Function 4. “Amplitude Shifts” (Stretches or Compressions) 5. Vertical Shifts

Sketching Transformations, cont. Step 2: Make a table that follows the order of operations for the function (Always start with the key points!) Step 3: Complete the table for the key points (0,,,, ) Step 4: Plot the key points Step 5: Connect the key points with a smooth line

Example 1: y = –sin(t) + 1 t

Domain: Range:

Example 2: y = 2 sin(t) – 3 t

Domain: Range:

Review – Solving for Coterminal Angles If an angle is negative or greater than 2π, then we add or subtract 2π until the angle is between 0 and 2π.  –5π + 2π = –3π + 2π = –π + 2π = π 

Your Turn: On a separate sheet of paper (or in the margin of your notes), find a coterminal angle between 0 and 2π for each of the following angles: π 3. 4π π

Problem 6: t

Domain: Range:

Problem 7: t