Consider the following function from Example 5. Work these exe...

Question

Consider the following function from Example 5. Work these exercises in order.

y = -2 - cot (x - π/4)

Use the fact that the period of this function is π to find the next positive x-intercept. Round to the nearest hundredth.

Accepted Answer

Welcome back everyone. In this problem, we want to find the next positive X intercept of the function given below, given that the least positive X intercept is 0.330. And we want to run our answer to the nearest 100. The function is Y equals negative five minus the cotangent of X minus 1/6 of pi. And for our answer choices A is 2.810. B is 3.470 C is 5.950 and the D is 1.240. Now, in this problem, let's try to visualize what's really going on here. So we have a cotangent graph and we're plotting a negative cotangent. OK? We don't, we don't necessarily have to try and plot it exactly how it looks. But it would be good for us to just to visualize, to get what's going on here, just an estimate. So what we're saying is that we have a code tangent graph, a negative quote tangent graph. It probably looks something like this. OK? And we are saying that the least positive intercept is 0.33. No, we are trying to find the next positive intercept. So in other words, when the function starts and goes again, what would be our positive intercept? That's what we're trying to figure out. Now notice that the distance between those positive intercepts or positive X intercepts would be the wavelength or the period of our function. So if we can find a period of our function here, then we should be able to tell what the next positive X intercept will be by adding the period to our least positive X intercept. OK. So let's try and figure that out. What do we know about the cotangent graph? Well, recall that every core tangent graph is written in the general form, Y equals a multiplied by the core tangent of B multiplied by X minus C OK plus D. OK. Where our period, in this case, since we're focusing on our period, our period would be equal to pi OK, equal to pi divided by the magnitude of the coefficient B where B is the coefficient of our X term here, our expression X minus C. So if we can figure out B from the equation we already have, then we can find a period and add it to 0.33. So for our function, OK, for our function, which is why equals, if we write it in the general form, it would be negative go a tangent of X minus 1/6 of pie minus five, I just rewrote it to more closely match the general form. Then now here, since this is an X term, this is one X, if I were to factor it out, it would be one multiplied by X minus 1/6 of pi. So this tells us, then OK, if we look at it here, this tells us then that B equals one. And if B equals one, that means the period is going to be equal to pi divided by one, which equals pi. So the period for our co tangent function is pi, all we need to do now is to add that to 0.33 to find the next positive Y intercept. Since we're running our answer to the nearest 100 let's assume. So assuming that pi equals 3.14 we write it to the nearest 100. OK? Then the next one I intercept the next positive way intercept would be equal to, for our exterm. It would be 0.33 plus 3.140 and 0.33 plus 3.14 equals 3.47. So the next positive Y intercept after 0.33 would be 3.470. When we look back on our answer choices that would have been answer choice. B thanks a lot for watching everyone. I hope this video helped.

Consider the following function from Example 5. Work these exercises in order.
y = -2 - cot (x - π/4)
Use the fact that the period of this function is π to find the next positive x-intercept. Round to the nearest hundredth.

Key Concepts

Cotangent Function

X-Intercepts

Periodicity in Trigonometric Functions

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