Determine the amplitude, the period, and the phase shift of...

Determine the amplitude, the period, and the phase shift of the function and, without a graphing calculator, sketch the graph of the function by hand. Then check the graph using
a graphing calculator.
$$y=\sin \left(\frac{1}{2} x\right)$$

Key Concepts

Horizontal Dilation

Horizontal dilation involves stretching or compressing the graph of a function along the x-axis. In trigonometric functions, this effect is induced by the coefficient of the variable inside the sine or cosine, which modifies the period and influences the spacing of key points on the graph.

Phase Shift

Phase shift refers to the horizontal displacement of the graph of a trigonometric function from its standard position. It is determined by a constant term added within the function's argument, resulting in a left or right shift of the entire waveform along the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions involves applying multiple transformations such as amplitude adjustments, period modifications, phase shifts, and sometimes vertical shifts to the basic sine or cosine function. This process includes identifying key points, understanding how the function is stretched or compressed, and accurately sketching the resulting wave by hand before verifying with a graphing tool.

Amplitude

Amplitude in a trigonometric function represents the vertical distance from the midline of the graph to its maximum (or minimum) point. It determines the height of the peaks and the depth of the troughs, reflecting the magnitude of the sine or cosine wave, regardless of any horizontal or vertical shifts.

Period

The period of a trigonometric function is the length over which the function repeats its pattern. In functions of the form y = sin(bx), the period is calculated by dividing the standard cycle length (2? for sine and cosine) by the absolute value of the coefficient b, indicating how the input is scaled horizontally.

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