How to Find the Equation of a Sine Wave from a Graph

To understand a sine wave, we first need to identify its key features.The midline is found by averaging the maximum and minimum y-values. Here, our maximum is 2 and minimum is negative 2, so the midline is at y equals zero.The amplitude is the distance from the midline to either the maximum or minimum point. In this case, our amplitude is 2 units.The period is the distance between two consecutive peaks or troughs. For this sine wave, one complete cycle spans two pi units.Let's summarize the key features we've identified in our sine wave.One complete cycle of our sine wave contains all these key features. It starts at the midline, reaches a maximum, returns to the midline, hits a minimum, and finally returns to its starting position.As we trace along the sine wave, notice how the y-value oscillates between the maximum and minimum, while maintaining the constant amplitude of 2 units from the midline.Remember, these key features - the midline, amplitude, and period - are essential for understanding and describing any sine wave.Now that we've identified the key features, let's convert our measurements into equation parameters.The first parameter is amplitude, represented by A. This is the vertical distance from the midline to either the maximum or minimum of the wave.The vertical shift, D, is the y-coordinate of the midline. This tells us how far up or down the entire wave is shifted from the x-axis.To find B, we use the formula B equals two pi divided by the period. When the period is two pi, B equals one.The phase shift C determines where the wave starts relative to the origin. If the wave is shifted right by pi over two, C equals negative pi over two in our equation.Putting our measurements together, we get y equals two sine of x minus pi over two plus one.These parameters will help us write our complete sine equation in the next section.Now that we have our parameters, let's assemble the complete sine equation.We'll substitute our values into the standard form equation y equals A sine of B x plus C plus D.Let's review what each parameter represents in our equation.This equation produces a sine wave with our specified features.Let's verify our equation by checking several key points along the curve.Now let's confirm that all key features of our wave match our intended values.The amplitude of 2 units, period of 2π, phase shift of π/2, and vertical shift of 1 unit all match our original specifications.