Math Grade 11, Sinusoidal functions Please state the period, amplitude, max/min values, range, domain, horizontal phase shift, and vertical displacement, of each of the above functions. please answer clearly and distinct the answer for each one, clearly. thanks.

Expert Answers

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The general form of a sinusoid is:


A: A is the amplitude. If A<0 the base function is reflected over the horizontal axis. (A is a vertical stretch/compression.)

B: B affects the period (it is a horizontal stretch/compression.) The period is `p=(2pi)/B` . If B<0 the base function is reflected over the y-axis.

h: h is a horizontal translation (phase shift.)

k: k is a vertical translation. It provides the midline for the sinusoid.

Given `y=-3cos(2x-30^@)+1`

Rewrite as `y=-3cos(2(x-15^@))+1`

The base function is the cosine function.

(1)The midline (vertical displacement is up 1) is y=1.

(2) The amplitude is 3

(3) The maximum value is 4 (3 units above the midline) while the minimum value is -2.

(4) The domain is all real numbers; the range is `-2<=y<-4`

(5) The horizontal phase shift is 15 units right. (`pi/12` units right)

(6) The function is reflected across the line y=1.

(7) The period is `p=(2pi)/2=pi` units.

The graph:

(Horizontal axis is in radians instead of degrees.)


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