Find the amplitude, period, phase displacement, and...

Find the amplitude, period, phase displacement, and sinusoidal axis location Without using your grapher, sketch the graph by locating critical points. Then check your graph using your grapher.
$$y=-8+10 \sin 5\left(\theta+6^{\circ}\right)$$

Key Concepts

Amplitude

The amplitude is the absolute value of the coefficient in front of the sine function. It represents the maximum vertical distance from the sinusoidal axis (or midline) to the peak of the wave, characterizing the strength or height of the oscillations.

Period

The period of a sinusoidal function is the horizontal length over which the function completes one full cycle. It can be determined from the frequency coefficient multiplying the input variable; specifically, the period is the length of the interval required for the sine function to repeat its pattern.

Phase Shift

The phase shift (or horizontal displacement) is the horizontal translation of the basic sine wave. It indicates how far and in which direction (left or right) the wave has been shifted. This shift is determined by the constant added to or subtracted from the variable in the function's argument.

Sinusoidal Axis

The sinusoidal axis, often referred to as the midline, is the horizontal line about which the sine wave oscillates. It corresponds to the constant vertical shift added to the function and serves as the baseline from which the amplitude is measured.

Critical Points

Critical points in the context of a sinusoidal function are key locations on the graph such as the maximum and minimum points, as well as the points where the function crosses the sinusoidal axis. These points are used to accurately sketch the graph by establishing a guide for the wave's behavior over one cycle.

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