Graph each function over a one-period interval.
y = -1 +...

Question

Graph each function over a one-period interval.

y = -1 + csc x

Accepted Answer

Graph. The following function considering only one period where our function is Y equals negative three plus cos of X, we are also given a coordinate plane to graph our function on. Now let's first look at the attributes of kin X cok and X is the reciprocal function, which means this is equivalent to one divided by the sine of X. So we know K seeking is reciprocal of sun. Now we can make use of the general form of sin X to graph our cosecant function. Our general form of sex is given by a sign of BX minus C plus D. Now we can look at our coin function to determine all these values. Let's transform our functions. First, we have negative three plus kokin X which can be transformed into negative three plus sign X to find our, our graph. Now let's focus on sin X. I will rewrite this as sin X minus three. To get our general form. We then notice our A is one RB is also one RC is zero because we don't have one. And finally, our D is negative three. This tells us a couple things. It tells us our amplitude is one RB is one which will use that in a second. And our D which is our vertical shift is negative three, which means this whole thing is shifted down by three. We can now find points on our functions X minus three to do this. We will need our period period is given by the equation two pi divided by B. Now we know our B is one, this gives us a period of two P. Now we can find X values for our graph. We don't have any horizontal ships that we noticed. So our first X value will be zero. Now we will make use of our period. Let's say we divide our period into five equal parts or rather five equal points. I should say which divides it into four equal parts to divide this, we will take two pi and divide it by four to get pi divided by two, which means our next point X two, we will say is zero plus our pi divided by two. This gives us a point on our function. Our next one, X three will take our previous value pi divided by two and add another pi divided by two that just gives us pie X four. We will take our previous answer pi in a pi divided by two. Again, this gives us three pi divided by two. Finally, our last point will be three pi divided by two plus pi divided by 22. Yet two P, we now have five X values to plot on a graph. Now, we just need to find our corresponding Y values. We'll make a table all of our values. Now, we need to find our Y values by plugging in X vs. We'll plug it into our sine function which we have sign X minus three, which we can say sign zero minus three or X equals zero. This gives us negative three. We can do the same. Now for every of the following points no, at pile divided by two, we would get sign pi divided by two which is one, subtracting three gives us negative two pi you once again get a negative three. For three pi divided by two, we would get negative four. And for two pi we get negative three, we now have all the points for our sine function which we can plot on our graph. We have zero, negative three pi divided by two, negative two pi negative three, three pi divided by two, negative four and two pi negative three. Now this gives us our sine function. However, we're looking for the cos secret function, we can make use of this graph to graph it. So in the cervical function, when sine of zero is equal to zero, we will have an Asymptote but we had two places where that happens or in this case, three places where it happened including our first point. If we look at our function. When S zero is equal to zero, we have an Asymptote, which means our first Asymptote happens Alex equals zero. Our second one happens that X equals pi and our final one happens at X equals two pi if we plugged all of those into our cos function, we all get undefined. No, our other points, we will draw some parabolas. Now at pi divided by two, we were at the maximum on the sine graph or at the peak of the sine graph, we will then draw pr from that peak that points upwards which travel reach on just a little bit. Let me get that right there and the same for the other value at three pi divided by two just pointing down because it's at a value we now have our co sequent graph N three plus Kosekin X. OK. I hope to help me solve the problem. Thank you for watching. Goodbye.

Graph each function over a one-period interval.
y = -1 + csc x

Key Concepts

Cosecant Function

Vertical Shifts

Periodicity of Trigonometric Functions

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