2. \( y=3 \sin (2 \pi x+4 \pi) \)

Did you know that the sine function is a wave on a graph? When you have an equation like \( y=3 \sin (2 \pi x+4 \pi) \), the 3 in front stretches the wave vertically, making it taller and more dramatic! Additionally, the \( 2 \pi x \) part of the equation indicates how quickly the wave oscillates horizontally; in this case, it completes a full cycle every 1 unit of x. Now, let's talk about how to apply this in real life! Sine waves can describe many phenomena, from sound waves in music to the ebb and flow of tides. If you were to graph this specific sine function, you'd see oscillations that repeat every 1 unit along the x-axis, perfect for understanding periodic behavior, like the rhythm of your favorite song or the cycles of day and night!