Below is a graph of the function y = sec ( b x − π ) y=\sec\left(bx-\pi\right) y = sec ( b x − π ) . Determine the value of b .

Let's give this problem a try. Here we're told below is a graph of the function y is equal to the secant of b x minus pi, determine the value of b. Now to solve this problem, when dealing with these kinds of trigonometric graphs, what I like to do is start with a trig function I'm more familiar with, and then work off of that one. Now what I can see from this is that we have a secant. And I know that secant is a reciprocal identity related to the cosine. So I'm first going to find the graph y is equal to the cosine of b x minus pi. Now, rather than trying to figure out what this graph exactly looks like by dealing with this kind of phase shift, and then moving the graph around, what I'm going to do is recognize that for this cosine function, it's going to have valleys At all of these places, we have these kind of downward curves, and it's going to have peaks at all these places we have upward curves. So the cosine function is gonna start at negative one, and it's going to go up and reach a peak at pi over two. Then it's gonna come down here and reach a valley at pi. We're going to peak again at three pi over two. We're going to valley down here at two pi, and then we're going to reach a peak again at five pi over two. So this is what the cosine function is going to look like when we graph it. Now, what I can do is I can recognize that this cosine function starts at a low value, and then it goes up, and then it goes back down, and it completes one full period when we get to pi on the x axis. So I can see that the period of this cosine function is pi. Now why do I need the period? Well that's because I know that the period of a cosine function is two pi over b. So what I can do is use this equation to find b by recognizing that the period is pi. So I'm going to set the period equal to pi, so we'll have that pi is equal to two pi over b, and then all I need to do from here is solve for b. Now I'm gonna do this by first multiplying b on both sides of the equation. That'll get the b's to cancel on the right, giving me pi times b is equal to two pi. Now from here what I can do is divide both sides of this equation by pi. And doing this will allow me to cancel the pi's on the left side and on the right side. So all I'm going to have left over is b is equal to two. So this right here is the solution for b, and the answer to this problem. So hope you found this video helpful. Thanks for watching.

Below is a graph of the function y = sec ⁡ ( b x − π ) y=\sec\left(bx-\pi\right) y = sec ( b x − π ) . Determine the value of b .

b = π 2 b=\frac{\pi}{2} b = 2 π

12. Trigonometric Functions

Graphs of Other Trigonometric Functions

Business Calculus

12. Trigonometric Functions

Graphs of Other Trigonometric Functions

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Multiple Choice

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

$b=2$

$b=4$

$b=\frac{\pi}{2}$

$b=\pi$

Verified step by step guidance

Step 1: Recall the general form of the secant function y = sec(bx - π). The parameter b affects the period of the function. The period of the secant function is given by T = (2π)/b.

Step 2: Analyze the graph provided. The graph shows vertical asymptotes at x = π/2, x = 3π/2, x = 5π/2, etc. These asymptotes occur at regular intervals, indicating the period of the function.

Step 3: Determine the period of the function from the graph. The distance between consecutive vertical asymptotes is π, which represents the period T of the function.

Step 4: Use the formula for the period of the secant function, T = (2π)/b, and substitute T = π. Solve for b: π = (2π)/b.

Step 5: Simplify the equation to find b. Multiply both sides by b and divide by π to isolate b, resulting in b = 2.

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Struggling with Business Calculus?

Below is a graph of the function y=sec(bx−π)y=\sec\left(bx-\pi\right)y=sec(bx−π). Determine the value of b.

Watch next

Graphs of Other Trigonometric Functions practice set

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