Graphing Form of Sine and Cosine Functions

The length of one cycle of a graph. Period The length of one cycle of a graph.

Requirements for a Sine/Cosine Graph x-intercept 2 Arrows (to show that there infinite cycles) 3 1 5 4 At least one Period (in other words, at least 5 consecutive critical points accurately plotted)

The Amplitude and the Effect of “a” Half of the distance between the maximum and minimum values of the range of a periodic function with a bounded range. a < 0 a > 1 a = 1 0<a<1 Amplitude = 1 1 0.5 3 The amplitude is the absolute value of a! It is a positive distance.

You need at least 5 consecutive critical points. Example: Sine Transformation: Flip the parent graph and translate it 3Pi/2 units to the left. Transformation: New Equation: y = 0 Period: x = -3π/2 You need at least 5 consecutive critical points.

You need at least 5 consecutive critical points. Example: Cosine Transformation: Translate the parent graph Pi/2 units to the left and 1 unit down. Transformation: New Equation: Period: y = -1 x = -π/2 You need at least 5 consecutive critical points.

Sine v Cosine Sine Cosine (Press the Graph)

Example: Sine or Cosine? Transformation: Amplitude - 2 Graph - Translation - 3 units up and … Period - 2π Orientation - New Equation: Since the Sine and Cosine graphs are periodic and translations of each other, there are infinite equations that represent the same curve. Here are two examples. y = 3

Example: Sine or Cosine? Transformation: Amplitude - 2 Graph - Sine Translation - 3 units up and 3π/4 to the left 3 units up and … Period - 2π Orientation - Positive New Equation: y = 3 x = -3π/4

OR

Example: Sine or Cosine? Transformation: Amplitude - 2 Graph - Cosine Translation - 3 units up and π/4 to the left 3 units up and … Period - 2π Orientation - Positive New Equation: y = 3 x = -π/4

Find the period for each graph and generalize the result. Changing the Period Find the period for each graph and generalize the result. 1 cycle in 2π 1/4 cycle in 2π Period = 2π Period = 8π 2 cycles in 2π 4 cycles in 2π Period = π Period = 0.5π

Determining the Period of Sine/Cosine Graph If or , the period (the length of one cycle) is determined by: Ex: What is the period of ?

Changing the Period w/o Affecting (h,k) The key point (h,k) is a point on the sine graph. Also, multiplying x by a constant changes the period. Below are two different ways to write a transformation. In order for the equation to be useful, it must directly change the graph in a specific manner. Which equation changes the period and contains the point (-3,4)? or

Graphing Form for Sine k h

Graphing Form for Cosine k h

Notation: Trigonometric Functions Correct way for the calculator! is equivalent to

You need at least 5 consecutive critical points. Example: Sine Transformation: Change the amplitude to 0.5 and the period to π. Then translate it π/2 units to the right and 1 unit down. Transformation: Not in Graphing form New Equation: Period: y = -1 x = π/2 You need at least 5 consecutive critical points.

You need at least 5 consecutive critical points. Example: Cosine Transformation: Change the period to 4π and translate the parent graph 1 unit up. Transformation: New Equation: y = 1 Period: x = 0 You need at least 5 consecutive critical points.

Example: Sine or Cosine? Transformation: Amplitude - 1.5 Graph - Translation - 2 units down and … Period - π/2 Orientation - New Equation: Since the Sine and Cosine graphs are periodic and translations of each other, there are infinite equations that represent the same curve. Here are two examples. y = -2 Period:

Example: Sine or Cosine? Transformation: Amplitude - 1.5 Graph - Cosine Translation - 2 units down and … 2 units down Period - π/2 Orientation - Positive x = 0 New Equation: y = -2 Period:

OR

Example: Sine or Cosine? Transformation: Amplitude - 1.5 Graph - Sine Translation - 2 units down and … 2 units down and 5π/8 to the right Period - π/2 Orientation - Negative x = 5π/8 New Equation: y = -2 Period: