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

Question

Graph each function over a one-period interval.

y = 1 - (1/2) csc (x - 3π/4)

Accepted Answer

Hello, everyone. We are asked to graph the following function considering only one period, we have Y equals two minus 1/4 kant of in parentheses, X minus pi divided by four. We provided a coordinate plane where the positive side of the X axis marked in pi divided by four increments. And the Y axis ranges from negative two to positive five with one unit increments labeled every two increments first to graph Koi Kant. I must first graph sign. So I'm gonna rewrite this as Y equals two minus 1/4 of the co of the, I'm sorry, the sign of X minus pi divided by four. And in this, I recall that our basic setup when we are graphing sign is that Y equals a multiplied by the sign in parentheses. BX minus C close parentheses plus D. So here A is gonna be negative 1/4 and this is how much it stretches or shrinks B helps us find the period. So B in this case, there's no number there. So B is one see is our horizontal shift and here it would be pi divided by four D is the vertical shift and that would be in this case, a positive too and we'll worry about that part when we get to the Koi camp. So first I want to find the period. So the period of sign, if it has nothing else is chew pi to find it, when we have values in front of the X, we'd have two pi divided by B B. In this case is 12 pi divided by one is two pi. So the period of this graph is two pi now to find what fits best for us, I'm gonna make a little XY table. And though I only need to do one period, I'm gonna start and kind of work from what I'm given on my graph. Meaning on my graph, I can start where X is zero. If X is zero and I plug that into our function, I find that I have a value of Y being the square root of two divided by eight, which is approximately 0.177. I'm gonna keep following the points that are on my graph because I think that'll make it easiest for us. So next, I'm going to use pi divided by four. And when I put pi divided by four into my sine function, I get a Y value of zero. Next pi divided by two. When that's placed into this Y sine function, we get a Y value of negative radical two divided by eight which is approximately negative 0.177. For our next value. Three pi divided by four. When that gets plugged in, we get a Y value of negative 14. Our next spot would be pie pi when plugged in gives us a negative radical two divided by eight. I should give myself a little more space here and that'll give us negative 0.177. Approximately five pie divided by four. We put that in. We end up with negative 1/4. Nope, I'm sorry. We end up with zero. Got ahead of myself. Next 13 pi divided by two. When we plug that in, we get the square root of two divided by eight which is approximately 0.177. Next we have seven pi divided by four which is a positive 1/4. And our final one for two pie two P will give us positive square root of two divided by eight which is approximately 0.177. So we got a little close to the edge there but I will make it fit give me one second. So that is again approximately 0.177. So now we'll put these points on our graph and we'll have the graph of the sign which will help us find the graph of the Koc can. So zero is plotted at Y for 0.177. It's very low. But roughly here pi divided by four is at zero on the Y pi divided by two is negative 0.177 pi divided or sorry. Three pi divided by four. Is that negative 1/4? Which is approximately there. It's a little further down than our previous value. Pie is at negative 0.177 five. Pi divided by four has a Y value of zero three. Pi divided by two has a Y value of 0.177 seven. Pi divided by four has a high value of 1/4 and two pi has a Y value of 0.177. So this is one full period of our curve. So let's connect that curve now from there, I can draw in asthma totes for Y equals zero. So this occurs at pi divided by four in the X. So I'm putting in a dash to line and at five pi divided by four on the X. So there's another vertical dashed line from there. I'm going to draw what our Ky can looks like before we do a vertical shift. So this means in the interval from zero to pi divided by four, we have the beginnings of a U shape but it kind of goes off our graph and we can't see it for the interval of pi divided by 4 to 5 pi divided by four, we have an upside down U shape that would share a point at three pi divided by four, negative 1/4. And then for our interval from five pi divided by 4 to 2. Pi we'd have a U shape that starts out taps into our cosine or sorry, our sine graph at seven pi divided by four and kind of just stops at the end of that period. So we have two kind of U shapes starting in opposite directions. And then we have an upside down U shape in the interval of four pi divided by 4 to 5 pi divided by four. Now we have a vertical shift of two. So this means all the points we just drew for our KC can actually are gonna go up two places. So in the interval from zero to pi divided by four, we need to go up to full blocks. So we're a little, we're probably about halfway, we're about 0.5. So we'll make this start now at 2.5 and continue upward. Next, we are at a three pi divided by four and negative 1/4. So we're gonna go up two. So that would be roughly three pi divided by four and um one and 3/4. So we'll make a downward U shape there. And for the interval of five pi divided by 4 to 2, pi we're going to go up to. So we have a point at approximately seven pi divided by four and two and a quarter, two and 1/4. And we'll make our U shape start there without crossing our asthma totes. So the black lines on my graph are our graph of Y equals two minus 1/4. The Koi Can of in parentheses X minus pi divided by four. Have a nice day.

Graph each function over a one-period interval.
y = 1 - (1/2) csc (x - 3π/4)

Key Concepts

Cosecant Function

Transformations of Functions

Period of Trigonometric Functions

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