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 4 pi. So that means that the period of this graph is 4 pi. So if the period is 4 pi, I can replace this period symbol with 4 pi, 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.

0. Functions

Graphs of Trigonometric Functions

Calculus

0. Functions

Graphs of Trigonometric Functions

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

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

Identify the period of the cotangent function from the graph. The function y = cot(bx + \frac{\pi}{2}) has vertical asymptotes at x = 2\pi, 6\pi, and 10\pi, indicating a period of 4\pi.

Recall that the period of the cotangent function y = cot(bx + \frac{\pi}{2}) is given by \frac{\pi}{b}.

Set the period \frac{\pi}{b} equal to the observed period from the graph, which is 4\pi.

Solve the equation \frac{\pi}{b} = 4\pi for b. This involves multiplying both sides by b and then dividing by 4\pi.

Conclude that the value of b is \frac{1}{4} after solving the equation.

5:53m

Watch next

Master Graph of Sine and Cosine Function with a bite sized video explanation from Nick

Start learning

Struggling with 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.

Watch next

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