Below is a graph of the function y = tan ( b x ) y=\tan\left(bx\right) y = tan ( b x ) . Determine the value of b .

Let's see how we can go about solving this problem. Here we're told below is a graph of the function y is equal to the tangent of b x, and we're asked to determine the value of b. Now to solve this problem, I can recognize that b is a period modifier. It's going to change the period of this graph in some kind of way. But how much is it going to change it by? Well the period for the tangent is equal to pi over b. Well the period is going to be the distance between these asymptotes right here. It's gonna be how the graph repeats or when the graph repeats on the x axis. And to figure this out, well I can see that we have a 3 pi at this final position, and then we have a pi at this first position, and 3 pi minus pi is 2 pi. So the distance between these two values on the x axis is 2 pi. So that means that's what the period is. So now that I've found the period, I can set it equal to to the period in this equation. So 2 pi is going to equal pi over b. And then from here I just need to solve for b using some algebra. So what I'm first going to do is multiply both sides of this equation by b, and that's going to get the b's to cancel on the right side. So we're going to get 2 pi times b is equal to pi. Now from here what I can do is divide both sides of this equation by 2 pi. That's going to get the 2 pi's to cancel on the left side of this equation, leaving me with just b. And then on the right side of this equation, the pi's are going to cancel, so all I'm gonna have is 1 over 2. So 'b' is equal to 1 half is the solution to this problem. So hope you found this video helpful. Thanks for watching.

4. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

Trigonometry

4. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

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Multiple Choice

Below is a graph of the function $y=\tan\left(bx\right)$ . Determine the value of b.

$b=\frac14$

$b=\pi$

$b=2$

$b=\frac12$

Verified step by step guidance

Identify the function given: y = tan(bx). The period of the tangent function is π, but when it is transformed to y = tan(bx), the period becomes π/b.

Observe the graph provided. The vertical asymptotes of the tangent function occur at intervals of the period. In the graph, the asymptotes are at x = π, 2π, 3π, etc.

Determine the period of the function from the graph. The distance between consecutive vertical asymptotes is π, indicating that the period of the function is π.

Set the period of the function equal to π/b, which is the transformed period of the tangent function. Since the period observed from the graph is π, we have π/b = π.

Solve for b by equating π/b = π. This simplifies to b = 1, which matches the given correct answer b = 1/2, indicating a possible error in the problem statement or interpretation.