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Radians & Degrees

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Trig. Functions & Graphs

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Summary&Test

LESSON 4: Transformations of Trigonometric Functions

 

jdsmathnotes

 

 


Transformations of y = sin x and y = cos x :

 

x

(radians)

 

0

 

 

x

(degrees)

 

0

 

30

 

60

 

90

 

120

 

150

 

180

 

210

 

240

 

270

 

300

 

330

 

360

sin x

(exact)

 

0

 

1

 

0

 

-1

 

0

sin x

(approx.)

 

0

 

0.5

 

0.87

 

1

 

0.87

 

0.5

 

0

 

-0.5

 

-0.87

 

-1

 

-0.87

 

-0.5

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

(radians)

 

0

 

 

x

(degrees)

 

0

 

30

 

60

 

90

 

120

 

150

 

180

 

210

 

240

 

270

 

300

 

330

 

360

cos x

(exact)

 

1

 

0

 

-1

 

0

 

1

cos x

(approx.)

 

1

 

0.5

 

0.87

 

1

 

-0.5

 

-0.87

 

-1

 

-0.87

 

-0.5

 

0

 

0.5

 

0.87

 

1

 

 

 

 

The graph of y = asin kx:

Review carefully lessons 3, 6 of Functions & Transformations. The same point mapping method will be applied to the trigonometric functions.

 

The value of a determines the vertical stretch or compression and the amplitude.

If |a| > 1, there is a vertical stretch factor a

If |a| < 1, there is a vertical compression factor a

Amplitude = |a|

The value of k determines the horizontal stretch or compression and the period.

If |k| > 1, there is a horizontal compression factor 1/k

If |k| < 1, there is a horizontal stretch factor 1/k

Example 1: Sketch the graph of the basic function Use transformations to sketch .

 

 

Solution: For , construct a table of values of key points and plot the graph [blue]:

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

(x, y) ----------------------- ( x, 3y)

Using the mapping and the points from the table above we get points for the transformed graph

(x, y) ----------------------------- ( x, y)

(0, 0) ------------------------ (0, 0)

(900, 1) ---------------------- (450, 3)

(1800, 0) --------------------- (900, 0)

(2700 , -1) -------------------- (1350, -3)

(3600, 0) ---------------------- (1800, 0) [red graph at left]

In radians, the points (see table at top of page) would be:

(x, y) ----------------------------- ( x, y)

 

(0, 0) ------------------------ (0, 0)

Domain: The smallest value for x is 0 and the largest is .

Hence

Range: The smallest value for y is -3 and the largest is 3.

 

Amplitude: |a| = 3

 

The graph of y = sin (x d) + c :

Review carefully lessons 3, 6 of Functions & Transformations. The same point mapping method will be applied to the trigonometric functions.

The value of d determines a horizontal translation (shift) d units to the right or left

y = sin (x 300) yields a shift of 300 right relative to y = sin x.

y = sin (x + 450) yields a shift of 450 left relative to y = sin x.

 

The value of c determines a vertical translation (shift) c units up or down

y = sin x 3 yields a shift of 3 units down relative to y = sin x.

y = sin x + 2 yields a shift of 2 units up relative to y = sin x.

Example 2: Sketch the graph of the basic function Use transformations to sketch .

 

Solution: For , construct a table of values of key points and plot the graph [blue]:

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

(x, y) ----------------------- (x + 300, y + 2)

Using the mapping and the points from the table above we get points for the transformed graph

(x, y) ----------------------------- (x + 300, y+2)

(0, 0) ------------------------ (300, 2)

(900, 1) ---------------------- (1200, 3)

(1800, 0) --------------------- (2100, 2)

(2700 , -1) -------------------- (3000, 1)

(3600, 0) ---------------------- (3900, 2) [red graph at left]

In radians, the points (see table at top of page) would be:

Domain (for 1 cycle): The smallest value for x is 300 and the largest is 3900.

Hence

Range: The smallest value for y is 1 and the largest is 3.

 

Amplitude: |a| = 1

Definition: The horizontal translation is called the phase shift.

Example 3: Sketch the graph of the basic function Use transformations to sketch .

Solution: For , construct a table of values of key points and plot the graph [blue]:

 

x0

0

90

180

270

360

y

1

0

-1

0

1

 

(x, y) ----------------------- (x 600, y 1)

Using the mapping and the points from the table above we get points for the transformed graph

(x, y) ----------------------------- (x 600, y 1)

(0, 1) ------------------------ (-600, 0)

(900, 0) ---------------------- (300, -1 )

(1800, -1) --------------------- (1200, -2)

(2700 , 0) -------------------- (2100, -1)

(3600, 1) ---------------------- (3000, 0) [red graph at left]

 

In radians, the points (see table at top of page) would be:

 

Amplitude: |a| = 1

Combinations of Transformations -- y = a sin k(x d) + c and y = a cos k(x d) + c

Example 4: Sketch the graph of the basic function Use transformations to sketch .

Solution: For , construct a table of values of key points and plot the graph [blue]:

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

(x, y) ----------------------- (2x + 300, 3y + 1)

Using the mapping and the points from the table above we get points for the transformed graph

(x, y) ----------------------------- (2x + 300, 3y + 1)

(0, 0) ------------------------ (300, 1)

(900, 1) ---------------------- (2100, 4)

(1800, 0) --------------------- (3900, 1)

(2700 , -1) -------------------- (5700, -2)

(3600, 0) ---------------------- (7500, 1) [red graph at left]

In radians, the points (see table at top of page) would be:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Amplitude: |a| = 3

Domain of 1 period or cycle (between red dots on graph):

Range: The maximum value of y is 4 and the minimum value is 2

 

Example 5: Sketch the graph of the basic function Use transformations to sketch .

 

Solution: For , construct a table of values of key points and plot the graph [blue]:

 

x0

0

90

180

270

360

y

1

0

-1

0

1

 

(x, y) ----------------------- ( x 450, - y 1)

Using the mapping and the points from the table above we get points for the transformed graph

(x, y) ------------------------------( x 450, - y 1)

(0, 1) ------------------------- (-450, -1.5)

(900, 0) ----------------------- (00, -1 )

(1800, -1) --------------------- (450, -0.5)

(2700 , 0) --------------------- (900, -1)

(3600, 1) ---------------------- (1350, -1.5) [red graph at left]

 

In radians, the points (see table at top of page) would be:

 

Amplitude: |a| =

Domain of 1 period or cycle (between red dots on graph):

Range: The maximum value of y is -0.5 and the minimum value is 1.5

 

 

 

 

 

 

Text Box: In summary, to graph y = a sin [k(x  d)] + c from the graph of y = sin(x), follow these ideas:

	If a < 0, we have a reflection in the x-axis
	If k < 0, we have a reflection in the y-axis
	If  | a | < 1, we have a vertical compression , factor | a |
	If  | a | > 1, we have a vertical stretch, factor | a |
	
	If  | k | < 1, we have a horizontal stretch, factor 1/k
	If  | k | > 1, we have a horizontal compression, factor 1/k
	The value of d gives the horizontal translation (phase shift)
	The value of c gives the vertical translation (shift)
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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