5.5 a&b Graphs of Sine and Cosine HW p. 589 1-23 odd, 31-65 odd

Graphical Analysis: graph the function in the given window Domain: Range: Continuous Alternately increasing and decreasing in periodic waves by Symmetry: Origin (odd function) Bounded Absolute Max. of 1 Absolute Min. of –1 No Horizontal Asymptotes No Vertical Asymptotes End Behavior: and do not exist

The “Do Now” – first, graph the function in the given window Other notes: This function is periodic, with period By definition, sin(t) is the y-coordinate of the point P on the unit circle to which the real number t gets wrapped by So now let’s “explore” where this wavy graph comes from…

Now, a Complete Analysis of the Cosine Function Domain: Range: Continuous Alternately increasing and decreasing in periodic waves by Symmetry: y-axis (even function) Bounded Absolute Max. of 1 Absolute Min. of –1 No Horizontal Asymptotes No Vertical Asymptotes End Behavior: and do not exist

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)

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!!!

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

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

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

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:

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 7. Horizontal shrink by 1/3 9. Period 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

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

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

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

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

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