Find an Equation for the Sine or Cosine Wave - Problem 1

I want to find the equation for another sine or cosine curve. I have one here and judging by the graph the fact that it starts on an x intercept I would say that this is a sine curve of some kind which means I’m going to have the equation y equals A times the sine of bx. It’s just a question of what A and b are.

Now because this curve goes downward instead or upward like sine normally does, I’m going to guess that there is a reflection across the x axis. In order to figure out what A is, I need to know that the amplitude of this is and it looks like it’s going to be 3/5. The coordinates to this point are the clue, the second coordinate the y coordinate of this point, 3/5. And if this distance is the same as this distance, then this will be negative 3/5 and that makes the amplitude 3/5.

So if I have a reflection across the x axis, and an amplitude of 3/5 that means A is negative 3/5, so y equals -3/5 sine of bx. I just need to find out what b is. The key to finding b is figuring out what the period of this function is. This point has an x coordinate of ¾ so well that’s ¾ all the way through a period, so this must be 1. If the period is 1 I use this formula, period equals 2 pi over b to find out what b is. I plug in 1 for the period and this one’s easy, I just multiply both sides by b and get b equals 2pi. That’s it, we’re done.

My final equation is y equals -3/5 sine of 2pi x. Let me move this guy over just a little bit, P. Y equals -3/5 sine of 2 pi x.