Using Transformations to Graph the Sine and Cosine Curves

• The following examples will demonstrate a quick method for graphing transformations of

sin cosy x y x

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Basic Function:

siny x

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Center Line:

3y

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Reflection

in x-axis?

yes

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Amplitude:

4 4

• Four units

above and below

the center line:

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Beginning and ending points of one period:

24

x

2 2 24

x

1 12 2

2 2 4x

8

x

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Beginning and ending points of one period:

8x

Length of one period

Beginning point

(phase shift)

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Beginning and ending points of one period:

8x

End point

Beginning point

(phase shift)

8 9

8 8 8

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Begin point

8

• End point

9

8

2

2

4

6

8

π

4

π

2

3π

4

π

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Divide the period into 4 intervals:

2

4 8

• From the first mark, use for succeeding marks. Because of the reflection, the marks will be low, centerline, high. The last mark will be centerline point, which is the end point that was already drawn.

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

2

2

4

6

8

π

4

π

2

3π

4

π

2

2

4

6

8

π

4

π

2

3π

4

π

Example 1:Use transformations to graph the following function:

3 4sin 24

y x

• Draw the curve

through the points.

2

2

4

6

8

π

4

π

2

3π

4

π

Example 2:Use transformations to graph the following function:

2 3cos 32

y x

• Basic Function:

cosy x

Example 2:Use transformations to graph the following function:

• Center Line:

2y

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Reflection

in x-axis?

yes

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Amplitude:

3 3

• Three units

above and below

the center line:

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Beginning and ending points of one period:

32

x

2 3 22

x

1 13 2

3 3 2x

2

6 3x

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Beginning and ending points of one period:

2

6 3x

Length of one period

Beginning point

(phase shift)

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Beginning and ending points of one period:

End point

4 3

6 6 6 2

2 3cos 32

y x

2

6 3x

Beginning point

(phase shift)

Example 2:Use transformations to graph the following function:

• Begin point

6

• End point

2

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Divide the period 2into 4 intervals:

2 1 2

3 4 12 6

• From the first mark, use for succeeding marks. Because of the reflection, the marks will be centerline, high, centerline. The last mark will be low, which is the end point that was already drawn.

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

2 3cos 32

y x

Example 2:Use transformations to graph the following function:

• Draw the curve through the points.

2 3cos 32

y x