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 two 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 two. So that means that the period for this sine wave is pi over two. 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 two, we'll replace this period here, is equal to two pi 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 two times b is equal to two pi. Now at this point, I can take both sides of the equation, and and I can multiply it by the reciprocal of pi over two, which would be two 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 two pi times two over pi. Well, this pi will cancel with that one, and two times two is four. So that means the solution for b is four. And that's how you can solve this problem.

12. Trigonometric Functions

Graphs of Sine & Cosine

Business Calculus

12. Trigonometric Functions

Graphs of Sine & Cosine

Struggling with Business Calculus?

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Multiple Choice

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

$b=\pi$

$b=2$

$b=\frac12$

$b=4$

Verified step by step guidance

Step 1: Recall the general form of the sine function y = sin(bx), where b affects the frequency of the wave. The frequency determines how many cycles the sine wave completes over a given interval.

Step 2: Identify the period of the sine wave from the graph. The period is the distance between two consecutive peaks or troughs. From the graph, observe that the period is π/2.

Step 3: Use the formula for the period of a sine function, which is given by Period = (2π)/b. Substitute the observed period (π/2) into the formula: π/2 = (2π)/b.

Step 4: Solve for b by isolating it in the equation. Multiply both sides of the equation by b and divide by π to find b = 4.

Step 5: Verify the solution by substituting b = 4 back into the formula for the period. The calculated period should match the observed period from the graph, confirming that b = 4 is correct.