Sinusoids and Transformations Sec. 4.4b

Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants and neither a nor b is 0. In general, any transformation of a sine function (or the graph of such a function – such as cosine) is a sinusoid. This is the format that we are used to seeing, thus it is OK to continue using this format…I use this format.

Transformations Horizontal stretches and shrinks affect the period There is a special vocabulary for describing our traditional graphical transformations when applied to sinusoids… Horizontal stretches and shrinks affect the period and the frequency. Vertical stretches and shrinks affect the amplitude. Horizontal translations bring about phase shifts.

Definition: Amplitude of a Sinusoid The amplitude of the sinusoid is Similarly, the amplitude of is Graphically, the amplitude is half the height of the wave.

Transformations Confirm these answers graphically!!! Find the amplitude of each function and use the language of transformations to describe how the graphs are related. (a) (b) (c) Amplitudes: (a) 1, (b) 1/2, (c) |–3| = 3 The graph of y is a vertical shrink of the graph of y by 1/2. 2 1 The graph of y is a vertical stretch of the graph of y by 3, and a reflection across the x-axis, performed in either order. 3 1 Confirm these answers graphically!!!

Definition: Period of a Sinusoid The period of the sinusoid is Similarly, the period of is Graphically, the period is the length of one full cycle of the wave.

Transformations Find the period of each function and use the language of transformations to describe how the graphs are related. Periods (a) (b) (c)

Transformations Find the period of each function and use the language of transformations to describe how the graphs are related. (a) The graph of y is a horizontal stretch of the graph of y by 3, a vertical stretch by 2, and a reflection across the x-axis, performed in any order. 2 1 (b) (c)

Transformations Confirm these answers graphically!!! Find the period of each function and use the language of transformations to describe how the graphs are related. (a) The graph of y is a horizontal shrink of the graph of y by 1/2, a vertical stretch by 3, and a reflection across the y-axis, performed in any order. 3 1 (b) Confirm these answers graphically!!! (c)

Definition: Frequency of a Sinusoid The frequency of the sinusoid is Reciprocal of the period!!! Similarly, the frequency of is Graphically, the frequency is the number of complete cycles the wave completes in a unit interval.

Transformations Find the frequency of the given function, and interpret its meaning graphically. Sketch the graph of the function in by Interpretation: Frequency: The graph completes 1 full cycle per interval of length . Period:

Transformations New Terminology: When applied to sinusoids, we How does the graph of differ from the graph of ?  A translation to the left by c units when c > 0 New Terminology: When applied to sinusoids, we say that the wave undergoes a phase shift of –c.

Transformations Confirm these answers graphically!!! Write the cosine function as a phase shift of the sine function. Write the sine function as a phase shift of the cosine function. Confirm these answers graphically!!!

Finally, a couple of whiteboard problems Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of the sine function. 1. Amplitude 2; Vertical stretch by 2 Amplitude 4; Vertical stretch by 4, Reflect across x-axis 3.

More whiteboard… Period ; Horizontal shrink 7. by 1/3 9. Find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of the cosine function. 7. Period ; Horizontal shrink by 1/3 9. Period ; Horizontal shrink by 1/7, Reflect across y-axis

Reminder: Graphs of Sinusoids The graphs of these functions have the following characteristics: Amplitude = Period = Frequency = A phase shift of A vertical translation of

Guided Practice by Graph one period of the given function by hand. Amplitude = Period = by

Guided Practice by Graph one period of the given function by hand. Amplitude = Period = by

Whiteboard by Graph three periods of the given function by hand. Amplitude = Period = by

Whiteboard by Graph three periods of the given function by hand. Amplitude = Period = by

Whiteboard by Graph three periods of the given function by hand. Amplitude = Period = by

Guided Practice Identify the maximum and minimum values and the zeros of the given function in the interval  no calculator! Maximum: At Minimum: At Zeros:

Whiteboard Identify the maximum and minimum values and the zeros of the given function in the interval  no calculator! Maximum: At Minimum: At and Zeros:

Whiteboard State the amplitude and period of the given sinusoid, and (relative to the basic function) the phase shift and vertical translation. Amplitude: Period: Phase Shift: Vertical Translation: 1 unit down

Whiteboard State the amplitude and period of the given sinusoid, and (relative to the basic function) the phase shift and vertical translation. Amplitude: Period: Phase Shift: Vertical Translation: 7 units up

Practice Problems Construct a sinusoid with period and amplitude 6 that goes through (2,0). First, solve for b: Find the amplitude: Let’s just take the positive value again. To pass through (2,0), we need a phase shift of 2  h = –2 Either will work!!!

Practice Problems Construct a sinusoid y = f(x) that rises from a minimum value of y = 5 at x = 0 to a maximum value of y = 25 at x = 32. First, sketch a graph of this sinusoid… Amplitude is half the height: The period is 64: We need a function whose minimum is at x = 0. We could shift the sine function horizontally, but it’s easier to simply reflect the cosine function……by letting a = –10

Practice Problems Support this answer graphically??? Construct a sinusoid y = f(x) that rises from a minimum value of y = 5 at x = 0 to a maximum value of y = 25 at x = 32. But since cosine is an even function: This function ranges from –10 to 10, but we need a function that ranges from 5 to 25………vertical translation by 15: Support this answer graphically???

Practice Problems One possibility: Construct a sinusoid with the given information. Amplitude: 2, Period: , Point (0,0) One possibility:

Practice Problems One possibility: Construct a sinusoid with the given information. Amplitude: 3.2, Period: , Point (5,0) One possibility: