Write the equation of the sine function in the form y=a...

Write the equation of the sine function in the form $y=a \sin b(x-c)+d$ given its characteristics. a) amplitude $4,$ period $\pi,$ phase shift $\frac{\pi}{2}$ to the right, vertical displacement 6 units down b) amplitude 0.5, period $4 \pi,$ phase shift $\frac{\pi}{6}$ to the left, vertical displacement 1 unit up c) amplitude $\frac{3}{4},$ period $720^{\circ},$ no phase shift, vertical displacement 5 units down

Write the equation of the sine function in the form $y=a \sin b(x-c)+d$ given its characteristics.
a) amplitude $4,$ period $\pi,$ phase shift $\frac{\pi}{2}$ to the right, vertical displacement 6 units down
b) amplitude 0.5, period $4 \pi,$ phase shift $\frac{\pi}{6}$ to the left, vertical displacement 1 unit up
c) amplitude $\frac{3}{4},$ period $720^{\circ},$ no phase shift, vertical displacement 5 units down

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Key Concepts

Vertical Displacement

Vertical displacement is the shift of the sine function graph up or down along the y-axis. This shift is represented by the parameter 'd' in the function y = a sin(b(x - c)) + d, where adding a positive 'd' moves the graph upward and a negative 'd' moves it downward.

Phase Shift

Phase shift is the horizontal translation of the sine function. It determines the left or right movement of the graph along the x-axis and is represented by the parameter 'c' in the function y = a sin(b(x - c)) + d. A positive phase shift moves the graph to the right, while a negative phase shift moves it to the left.

Amplitude

Amplitude refers to the distance from the midline of the sinusoidal graph to its peak (or trough). It represents the vertical stretch or compression of the sine function and is given by the absolute value of the coefficient 'a' in the function y = a sin(b(x - c)) + d.

Period

The period of a sine function is the length of one complete cycle on the x-axis. It determines how frequently the sine wave repeats its pattern. In the standard sine function, the period is adjusted by the factor 'b', and is given by the formula period = (2?)/|b| for radian measures.

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