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 2, 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 2 of these asymptotes. So this would be the distance right here. Now looking at this distance I see that we finish at 6 pi, and that we started at 2 pi. So the difference between 6 pi and 2 pi is 4pi. So that means that the period of this graph is 4pi. So if period is 4pi, I can replace this period symbol with 4pi, and that's going to equal pi over b. So all I need to do from here is solve for b. Now what I'm first going to do is multiply both sides of this equation by 1 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 4 is equal to 1 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 4 b is equal to 1. And then I just need to divide both sides of this equation by 4. This will get the 4s to cancel on the left side, leaving me with just b is equal to one 4th. So 1 4th is the solution for b and the answer to this problem. Hope you found this video helpful.

10. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

Precalculus

10. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

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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.

$b=$\frac$14$

$b=1$

$b=2$

$b=$\frac$12$

Verified step by step guidance

The function given is y = cot(bx + $\frac{\pi}{2}$). The cotangent function has vertical asymptotes where its argument is an odd multiple of $\frac{\pi}{2}$.

Identify the vertical asymptotes from the graph. They occur at x = 2$\pi$, 4$\pi$, 6$\pi$, 8$\pi$, and 10$\pi$.

The period of the cotangent function is the distance between consecutive vertical asymptotes. From the graph, the period is 2$\pi$.

The period of the function y = cot(bx + $\frac{\pi}{2}$) is given by $\frac{\pi}{b}$. Set this equal to the observed period: $\frac{\pi}{b}$ = 2$\pi$.

Solve for b by equating $\frac{\pi}{b}$ = 2$\pi$. This gives b = $\frac{1}{2}$.

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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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