Below is a graph of the function y = cot ( b x + π 2 ) y=\cot\left(bx+\frac{\pi}{2}\right) y = cot ( b x + 2 π ) . Determine the value of b .

Let's try solving this problem. Here we are told below is a graph of the function y is equal to the cotangent of bx plus pi over two. We're asked to determine the value of b. Now, to solve this problem, what I'm going to do is recognize something about the cotangent. The cotangent has a period that repeats at pi over b units. Now this tells me that all I really need to do is figure out the period of this cotangent graph, and that is going to tell me how to find the b value. Now looking at this graph, I can see that the period is where this cotangent repeats and it's going to be between two of these asymptotes. So this would be the distance right here. Now looking at this distance, I see that we finish at six pi and that we started at two pi. So the difference between six pi and two pi is four pi. So that means that the period of this graph is four pi. So if the period is four pi, I can replace this period symbol with four pi, and that's going to equal pi over b. So all I need to do from here is solve for b. Now Now what I'm first going to do is multiply both sides of this equation by one over pi. And the reason I'm doing this is because this will allow me to cancel the pi's on the left side and on the right side. So what I'm going to end up with is four is equal to one over b. Now what I can do is multiply b on both sides of this equation. This will get the b's to cancel here, giving me that four b is equal to one. And then I just need to divide both sides of this equation by four. This will get the fours to cancel on the left side, leaving me with just b is equal to one fourth. So one fourth is the solution for b and the answer to this problem. Hope you found this video helpful.

12. Trigonometric Functions

Graphs of Other Trigonometric Functions

Business Calculus

12. Trigonometric Functions

Graphs of Other Trigonometric Functions

Struggling with Business Calculus?

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Multiple Choice

Below is a graph of the function $y=\cot\left(bx+\frac{\pi}{2}\right)$ . Determine the value of b.

$b=\frac14$

$b=1$

$b=2$

$b=\frac12$

Verified step by step guidance

Step 1: Recall the general form of the cotangent function. The function given is y = cot(bx + π/2). The period of the cotangent function is determined by the coefficient b, and the formula for the period is T = π / b.

Step 2: Analyze the graph. From the graph, observe that the cotangent function repeats its pattern every 4π units. This means the period T of the function is 4π.

Step 3: Set up the equation for the period. Using the formula T = π / b, substitute the observed period T = 4π into the equation: 4π = π / b.

Step 4: Solve for b. To isolate b, multiply both sides of the equation by b and then divide by 4π. This will give b = 1/4.

Step 5: Verify the solution. Substitute b = 1/4 back into the period formula T = π / b to confirm that the period is indeed 4π, matching the graph.

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Struggling with Business Calculus?

Below is a graph of the function y=cot(bx+π2)y=\cot\left(bx+\frac{\pi}{2}\right)y=cot(bx+2π). Determine the value of b.

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Graphs of Other Trigonometric Functions practice set

Below is a graph of the function $y=\cot\left(bx+\frac{\pi}{2}\right)$ . Determine the value of b.