explain in trig Amplitude, midline, and period – sinusoidal functions

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Welcome to our exploration of sinusoidal functions. These wave-like functions, such as sine and cosine, have three key characteristics that define their shape and position on the coordinate plane. These are amplitude, midline, and period. In this series, we'll examine each of these components in detail to understand how they affect the graph of a sinusoidal function. Let's start with amplitude. The amplitude of a sinusoidal function is the maximum distance from the midline to either the highest or lowest point of the wave. It represents how 'tall' the wave is. Mathematically, amplitude equals half the difference between the maximum and minimum values of the function. For functions in the form y equals A sine of x or y equals A cosine of x, the amplitude is simply the absolute value of A. In our example, y equals 2 sine of x has an amplitude of 2, meaning the wave extends 2 units above and below the midline. Next, let's examine the midline of a sinusoidal function. The midline is the horizontal line that the function oscillates around. It represents the vertical shift of the wave from the x-axis. Mathematically, the midline equals the average of the maximum and minimum values of the function. For functions in the form y equals A sine of x plus D or y equals A cosine of x plus D, the midline is simply at y equals D. In our example, y equals 1 plus sine of x has a midline at y equals 1, meaning the entire wave is shifted up 1 unit from the x-axis. The function oscillates between a maximum value of 2 and a minimum value of 0, with the midline exactly in the middle at y equals 1. Finally, let's explore the period of a sinusoidal function. The period is the horizontal distance required for one complete cycle of the wave to repeat itself. It represents how 'long' the wave is before it starts repeating the same pattern. For functions in the form y equals A sine of B x or y equals A cosine of B x, the period is 2π divided by the absolute value of B. The standard sine function, y equals sine of x, has a period of 2π, meaning it completes one full cycle every 2π units along the x-axis. When we compress the function horizontally, as in y equals sine of 2x, the period becomes π, which is half the original period. This means the compressed function completes its cycle twice as fast as the standard function. To summarize what we've learned about sinusoidal functions: The amplitude is the maximum distance from the midline to the peak or trough, representing the wave's height. The midline is the horizontal line the function oscillates around, representing the vertical shift. The period is the horizontal distance required for one complete cycle, representing the wave's length. For functions in the general form y equals A sine of B x plus C plus D or y equals A cosine of B x plus C plus D: the amplitude equals the absolute value of A, the midline is at y equals D, and the period equals 2π divided by the absolute value of B. Understanding these three key components allows us to analyze and graph any sinusoidal function.