Graphs Transformation of Sine and Cosine Consider the form y = A sin (Bx – C) + D and y = A cos (Bx – C) + D where A, B, C, and D are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs.

Vertical Shift Let’s observe the effect of the constant D.

Vertical Shift

The Constant D The constant D in y = A sin (Bx – C) + D and y = A cos (Bx – C) + D translates the graphs up D units if D > 0 or down |D| units if D < 0.

The Amplitude The amplitude of the graphs of Let’s observe the effect of the constant A.

The Amplitude

The Constant |A| is the amplitude of the graph If |A| > 1, then there will be a vertical stretching. If |A| < 1, then there will be a vertical shrinking. If A < 0, the graph is also reflected across the x-axis.

The Constant B Let’s observe the effect of the constant B.

The Constant B

The Constant B

The Constant B

Copyright © 2009 Pearson Education, Inc. The Constant B If |B| < 1, then there will be a horizontal stretching. If |B| > 1, then there will be a horizontal shrinking. If B < 0, the graph is also reflected across the y-axis. Copyright © 2009 Pearson Education, Inc.

Period The period of the graphs of y = A sin (Bx – C) + D and y = A cos (Bx – C) + D is

Period The period of the graphs of y = A csc (Bx – C) + D and y = A sec (Bx – C) + D is

Period The period of the graphs of y = A tan (Bx – C) + D and y = A cot (Bx – C) + D is

The Constant C Let’s observe the effect of the constant C.

The Constant C

The Constant C

The Constant C

The Constant C If B = 1, then if |C| < 0, then there will be a horizontal translation of |C| units to the right, and if |C| > 0, then there will be a horizontal translation of |C| units to the left.

Combined Transformations It is helpful to rewrite y = A sin (Bx – C) + D and y = A cos (Bx – C) + D as and

Phase Shift The phase shift of the graphs and is the quantity

Phase Shift If C/B > 0, the graph is translated to the right |C/B| units. If C/B < 0, the graph is translated to the right |C/B| units.

Transformations of Sine and Cosine Functions To graph and follow the steps listed below in the order in which they are listed.

Transformations of Sine and Cosine Functions 1. Stretch or shrink the graph horizontally according to B. |B| < 1 Stretch horizontally |B| > 1 Shrink horizontally B < 0 Reflect across the y-axis The period is

Transformations of Sine and Cosine Functions 2. Stretch or shrink the graph vertically according to A. |A| < 1 Shrink vertically |A| > 1 Stretch vertically A < 0 Reflect across the x-axis The amplitude is A.

Transformations of Sine and Cosine Functions 3. Translate the graph horizontally according to C/B. The phase shift is

Transformations of Sine and Cosine Functions 4. Translate the graph vertically according to D. D < 0 |D| units down D > 0 D units up

Example Sketch the graph of Find the amplitude, the period, and the phase shift. Solution:

Example Solution continued To create the final graph, we begin with the basic sine curve, y = sin x. Then we sketch graphs of each of the following equations in sequence.

Example Solution continued

Example Solution continued

Example Solution continued

Example Solution continued

Example Solution continued