Warm UP 1) Name the following parent graph: 2) Where is a point of inflection(s) for the function y=cos(x) on the interval [0 o, 360 o ]? 3) On what subinterval(s) is y=sin(x) increasing?

Exploration: Transformed Periodic Functions

General Sinusoidal Equation is the amplitude B is the reciprocal of the horizontal dilation (period) K is the location of the sinusoidal axis (vertical shift) H is the phase shift (horizontal translation)

Period Period = Frequency =

Transformations Amplitude (A) – makes the graph shorter or taller Period (B) – how fast the wave is going Phase Shift (H)– horizontal shift; left or right (opposite!) Phase Displacement (K)– vertical shift; up or down (same!)

Example 1 Let y = 4 sin x. What transformation of the parent sine function is this? How would the transformed graph compare to the graph of the parent function? Check your guess by graphing both the parent sinusoid and the transformed sinusoid on the same screen.

Example 2 Let y = 2 sin (x – 45). What transformation of the parent sine function is this? How would the transformed graph compare to the graph of the parent function? Check your guess by graphing both the parent sinusoid and the transformed sinusoid on the same screen.

Example 3 Let y = cos(2x – 180) + 1. What transformation of the parent sine function is this? How would the transformed graph compare to the graph of the parent function? Check your guess by graphing both the parent sinusoid and the transformed sinusoid on the same screen.

Example 4 Let y = tan (x + 90) – 2. What transformation of the parent sine function is this? How would the transformed graph compare to the graph of the parent function? Check your guess by graphing both the parent sinusoid and the transformed sinusoid on the same screen.

Example 5 Let. What transformation of the parent sine function is this? How would the transformed graph compare to the graph of the parent function? Check your guess by graphing both the parent sinusoid and the transformed sinusoid on the same screen.

Graphing Practice Example 6 Graph one period of f(x) = - cos (3x)

Transformation Example 7 Graph one period of f(x) = tan(x) – 1

Transformation Example 8 Graph one period of f(x) = sin (2x + 180) + 3 starting with the phase shift.

Example 9 Suppose that a sinusoid has period 12° per cycle, amplitude 7 units, phase displacement -4° with respect to the parent cosine function, and a sinusoidal axis 5 units below the Θ-axis. Without using your graphing calculator, sketch this sinusoid and then find an equation for it. Verify with your calculator that your equation and the sinusoid you sketched agree with each other.

Example 9 Continued… First draw the sinusoidal axis at y = -5. Use the amplitude, 7, to draw the upper and lower bounds 7 units above and 7 units below the sinusoidal axis.

Example 9 Continued… Next find some critical points on the graph. Start at Θ = -4°, because that is the phase displacement, and mark a high point on the upper bound. (Cosine) Then use the period, 12°, to plot the ends of the next two cycles.

Example 9 Continued… Mark some low critical points halfway between consecutive high points. Now mark the points of inflection. They lie on the sinusoidal axis, halfway between consecutive high and low points. Finally, sketch the graph by connecting critical points and points of inflection with a smooth curve.

Example 9 Continued… Because the period of this sinusoid is 12° and the period of the parent cosine function is 360°, the horizontal dilation is 1/30. The coefficient B is the reciprocal of 1/30, namely, 30.