The Sine Graph: Introduction and Transformations

26 April 2011

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

2

23

2

0

Key Features of y = sin(t)

Maximum:

Minimum:

Domain:

Range:

2

2

– 2

0

Multiple Revolutionst sin(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

20

2

–21 Period 1 Period

Period: π

Periodicity, cont.

2

tsiny )t4sin(y

2

2

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.

|b|

2

2

tsin)t(f

4

t3sin)t(f

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.

5

t2sin6)t(f

8

tsin4)t(f

4

tsin)t(f

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 Examples1. f(t) = 6sin(4t) 2. f(t) = –5sin(6t)

3. 4.)tsin(3

2)t(f

3

tsin

5

1)t(f

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

2

23

2

t f(t) = sin(t) f(t) = sin(t) + 3

0

Refection Questions3. What transformations did you see?

4. A.

B.

5. A.

B.

Transformationsf(t) = a sin(bt – c) + k

Vertical Shift

Pay attention to the parentheses!!!

Investigation #2!

2

23

2

t f(t) = sin(t) f(t) = 2sin(t)

0

Reflection Questions4. What transformation did you see?

Stretch = coefficient is a whole # Compression = coefficient is a fraction5. A.

B.C.

6. A.B.C.

Transformationsf(t) = a sin(bt – c) + k

Vertical Shift

Stretch or Compression

“Amplitude Shift”

Pay attention to the parentheses!!!

Reflection Questions4. What transformation did you see?

5. A.

B.

C.

6. A.

B.

C.

Transformationsf(t) = a sin(bt – c) + k

Vertical Shift

Stretch or Compression

“Amplitude Shift”

Pay attention to the parentheses!!!

Period Shift

Reflection Questions4. What transformation did you see?

4. A.

B.

C.

6. A.

B.

C.

Transformationsf(t) = a sin(bt – c) + k

Vertical Shift

Stretch or Compression

“Amplitude Shift”

Pay attention to the parentheses!!!

Period ShiftPhase Shift

Identifying Transformationsf(t) = 2 sin(4t – π) – 3

“Amplitude Shift”:

Period Shift:

Phase Shift:

Vertical Shift:

“Amplitude Shift”:

Period Shift:

Phase Shift:

Vertical Shift:

623

tsin

3

1)t(f

Your Turn: Identify the transformations of the following

sine graphs:

1. f(t) = 3 sin(t) + 2 2. f(t) = –sin(t – 4) + 1

3. 4.7)t2sin(3

1)t(f 8

5

t2sin

3

2)t(f

Sketching Transformations Step 1: Identify the correct order of

operations for the function1. 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

2 2

3 2

Example 1: y = –sin(t) + 1

t

0

2

2

3

2

Example 1: y = –sin(t) + 1

Domain:

Range:

Example 2: y = 2 sin(t) – 3

t

0

2

2

3

2

Example 2: y = 2 sin(t) – 3

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

2

32

2

7

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:

1. 2. 3π 3. 4π

4. 5. 3π

2

2

3

Problem 6:

t

0

2

2

3

2

2

tsiny

Problem 6:

Domain:

Range:

2

tsiny

Problem 7:

t

0

2

2

3

2

tsin6y