In the graphing formula y = A sin[B(x - c)] + D, what does |A| represent?

In the graphing formula ( y = A \sin[B(x - c)] + D ), the term ( |A| ) represents the amplitude of the graph. Amplitude refers to the distance from the midline of the graph (which is typically the line ( y = D )) to the maximum or minimum value of the sine function.

For a sine wave, the amplitude dictates how far the wave oscillates above and below its equilibrium position. Since the sine function itself oscillates between -1 and 1, multiplying it by ( A ) stretches the graph vertically. Thus, if ( A ) is positive, the graph will reach a maximum value of ( D + |A| ) and a minimum value of ( D - |A| ). Therefore, the magnitude of ( A ) determines the height of the wave's peaks and the depths of its troughs, defining the overall "size" of the oscillation.

This understanding is critical in the analysis of periodic functions because it helps to predict the behavior of the sine wave based on its amplitude, without being confused with other parameters such as period, vertical shift, or horizontal shift, which are represented by other components in the formula.