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.
$$y=-10+20 \sin \frac{1}{2}\left(\theta-120^{\circ}\right.$$

Key Concepts

Graphing and Critical Points

Graphing a sinusoidal function involves applying the transformations in sequence to the basic sine curve. Critical points, such as the maximum, minimum, and midline intersections, can be determined by considering the amplitude, period, phase shift, and vertical shift. These points guide the sketching of the complete cycle of the sine graph, ensuring an accurate representation of its oscillatory behavior.

Sinusoidal Axis (Vertical Shift)

The sinusoidal axis, also known as the midline, is the horizontal line about which the sine function oscillates. In a function of the form y = D + A sin(B(? - C)), the value D provides the vertical shift, positioning the midline at y = D, and effectively shifting the entire graph up or down.

Amplitude

Amplitude represents the height of the wave from its sinusoidal axis to its maximum or minimum value. In a transformed sine function of the form y = D + A sin(B(? - C)), the amplitude is given by |A|, which determines the vertical stretching or compressing of the graph.

Period

The period of a sine function is the length of one complete cycle of the wave. When a sine function is written as y = A sin(B(? - C)), the period is calculated as 360°/|B| (when working in degrees) or 2?/|B| (when working in radians). This value indicates how long it takes for the function to repeat its pattern.

Phase Displacement (Phase Shift)

Phase displacement, or phase shift, is the horizontal translation of the sine graph. In the function y = A sin(B(? - C)), the phase shift is the constant C, which moves the starting point of the cycle to the right if C is positive (and to the left if negative). This shift affects where the sine wave begins its cycle along the horizontal axis.

Transcript

Please give Ace some feedback