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Sinusoids

Sinusoids. You constructed the graphs of y =sin x and y =cos x in a previous activity. These graphs have a wave-like structure and are called sinusoids . The graph of a sinusoid is periodic , meaning that it repeats itself infinitely (i.e., as we go around the unit circle). y =sin x.

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Sinusoids

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  1. Sinusoids • You constructed the graphs of y=sin x and y=cos x in a previous activity. • These graphs have a wave-like structure and are called sinusoids. • The graph of a sinusoid is periodic, meaning that it repeats itself infinitely (i.e., as we go around the unit circle).

  2. y=sin x

  3. Terminology • y=sin x and y=cos x are parent functions. • The period tells us how often the graph repeats itself. • The amplitude tells us about the vertical height of the graph: it is half of the range.

  4. y=cos x Domain: all reals Range: -1 ≤ y ≤ 1 Amplitude: 1 Period: 2π 1 -2π -π 0 π 2π -1

  5. Transformations • We can alter the graph of the parent function in multiple ways. These parameter changes create functions that are transformations of the originals. • All sinusoids can be written in the form: y=Asin[B(x-C)]+D

  6. Transformations y=Asin[B(x-C)]+D • A affects the amplitude • The graph is stretched vertically by a factor of |A|. • If A is negative, it mirrors the graph across the x-axis. y=sin x y=3sin x

  7. Transformations y=Asin[B(x-C)]+D • D is a vertical shift. • The graph is translated vertically by D units. y=sin x y=sin x + 2

  8. Transformations y=Asin[B(x-C)]+D • C is a horizontal shift known as a phase shift. • The graph moves to the right C units. y=sin x y=sin (x - π/6)

  9. Transformations y=Asin[B(x-C)]+D • B alters the period of the graph. • If B>1, the function “repeats” more often. • The new period can be calculated using the formula 2π/B. y=sin x y=sin 3x

  10. Properties of Transformed Sinusoids y=3sin[2(x+(π/5))]-2 • The new period is (2π/2) = π. • The amplitude of 3 stretches the graph and makes the range -3 ≤ y ≤ 3… • … and the vertical shift then moves the graph down to -5 ≤ y ≤ 1. • The phase shift moves the graph to the left (π/5) units.

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