Given below is the graph of the function y = sin ⁡ ( b x ) y=\sin\left(bx\right) y = sin ( b x ) . Determine the correct value for b.

Hey everyone. Let's see how we can solve this problem. Here we're told given below is the graph of the function y is equal to the sine of b x. Determine the correct value for b. Now I want you to recognize that this value b is going to change the period of our sine function in some kind of way. And recall that for sine and cosine functions, the period is equal to 2 pi divided by b. Now what I'm going to do is see if using this graph, I can determine the period of this sine function. And the way that I can do this is by figuring out the distance on the x axis that it takes for this wave to complete one full cycle. Well, if I start here at the origin for our wave, I can go up and then down, and then start going back up again when we get to pi over 2. So that means that the period for this sine wave is pi over 2. And what that tells me is I can take this period here, and I can set it equal to the period in that equation, and just solve for b using algebra. So we'll have that pi over 2, we'll replace this period here, is equal to 2pi over b. Now at this point all I need to do is solve for b. Now to do this I'm first going to take both sides of the equation and multiply it by b, and then I'll get the b's to cancel on the right. Giving me pi over 2 times b is equal to 2pi. Now at this point, I can take both sides of the equation, and I can multiply it by the reciprocal of pi over 2, which would be 2 over pi. Because doing this allows me to clear the fraction on the left side of the equation, because the twos are going to cancel and the pis are going to cancel. So all I'll be left with is b on the left side, then b is going to equal 2 pi times 2 over pi. Well, this pi will cancel with that 1, and 2 times 2 is 4. So that means the solution for b is 4, and that's how you can solve this problem.