Graph each function over a one-period interval.
y = -½ c...

Question

Graph each function over a one-period interval.

y = -½ cos (πx - π)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function consider only one period. Our function is Y equals negative four thirds cosine of. And then all in parentheses pi X divided by two minus pi divided by three. Then we're given a blank graph. We have a vertical Y axis, a horizontal X axis. They come together at the origin. The domain for what's drawn for our X axis is from negative one to positive five that's marked off in increments of one. And the range for what's drawn for our Y axis is from negative two to positive two, which is marked off in increments of one half. All right. So we're given a cosine function cosine function is traditionally given to us in the format of Y equals C plus A multiplied by the cosine of then open parentheses. B multiplied by the quantity of X minus D close both sets of parentheses. So we can compare our given function with that formula and identify our ABC and D and that can help us graph our function. One thing to note ours doesn't completely line up. There's one thing that's different here. So first let's look at the C term, C term is what's added to our cosine function, there's nothing being added or subtracted. So our C term is zero, C indicates a vertical shift. And because C equals zero, there is none A is being multiplied by our cosine function. Here, we can see that A is negative four thirds. A has to do with amplitude and reflection over the X axis. And so for amplitude, we recall from previous lessons that that is going to be the absolute value of A A is negative four thirds. So it's going to be the absolute value of negative four thirds, which is a positive four thirds. So that's our amplitude. And because our, a term is negative, it is going to indicate reflection over the X axis. Now our B term, that's the thing that's different. We can't just look and match and see what our B term is because we have something being directly multiplied by our X. It hasn't been factored out of both the term with the X and the one being subtracted from it. So we need to factor out what's being multiplied by our X. We can see that that's going to be pi divided by two. So that is going to be our B but then what's our D term going to be? So we have X minus and we're taking pi divided by three, divided by pi divided by two. We're factoring out the pi divided by two. So that, that's our B term and pi divided by three, divided by pi divided by two, is two thirds. So now we can see our B term equals pi divided by two and our D term, what's being subtracted from the X is two thirds? Now let's talk about what each of those means. The B term has to do with the period of our function. We recall from previous lessons that the cosines period is given to us as two pi divided by B. So here our B is pi divided by two. So we have two pi divided by pi divided by two. And that is going to simplify to be four. So we have a period of four. Now for our D term that indicates a phase shift. And because our D is equal to a positive two thirds, it's going to shift to the right two thirds of a unit. All right. So if you can figure out, you know what cosine looks like, you can do all of those transformations in your mind. That's fantastic. But another thing we can do at this point is we can make an XY table, we can identify our five key X values. Plug those into our original function, get the corresponding Y values and then plot the points and connect them, then we'll have our cosine function. So for our first key X value, typically we would start right at zero for X, but we have a phase shift, it is a phase shift to the right two thirds of a unit. So we're going to take zero plus two thirds and that's going to be two thirds for our first X value. Now, for the other four X values, we're going to add 1/4 of our period, which is four, so 1/4 of four equals one. So we're just adding one. So two thirds plus one is five thirds, five thirds plus one is eight thirds, eight thirds plus one is 11 3rd and 11 3rd plus one is 14 thirds. Those are our five key X values. Now, we just plug those in to our function. And I'll just show you that for the first one. So for an X value of two thirds, we'll put into the calculator negative four thirds multiplied by the cosine of, then it gives you parentheses and then you have pi multiplied by our X value, which is two thirds. If you can put fractions into your calculator, that's great. Otherwise, just put in parentheses, two, divided by three, close the parentheses and then that's divided by two in the denominator. And then that fraction is minus pi divided by three, close those parentheses. When you enter all of that into the calculator, you'll get a Y value of negative four thirds, which makes sense. We had no vertical shift and our amplitude is four thirds and it was reflected over the X axis. So if you get negative four thirds, that makes sense. All right, plugging in five thirds for X and if you need to pause the video and do that, go for it. But if you plug that in, you'll get a Y value of zero for an X value of eight thirds, you'll get a corresponding Y value of four thirds for an X value of 11 3rd, you'll get a corresponding Y value of zero. And for an X value of 14 3rd, you'll get a corresponding Y value of negative four thirds. Now just plotting all of these points. So for X equals two thirds from the origin, we go to the right two thirds of a unit and then we're going down negative or we're going down four thirds of a unit which is about one and a third. And remember that this is marked off in halves. So it's approximately here. All right, the next point is five thirds zero. So one unit to the right of our previous X value. And then on the X axis, next point is eight thirds, four thirds. So one unit to the right of our previous unit and then up four thirds which is one and a third, next 0.11 3rd zero. So one unit to the right of our previous unit, but back down on the X axis and then finally 14 3rd negative four thirds. So one unit to the right and then one and a third units below the X axis. So we can connect these as smoothly as possible. We know that previous to a 0.2 3rd, negative four thirds, that's a minimum. So to the left of that at the Y axis, it's coming, it's heading down from the Y axis to intersect that point and then it'll come up and pass through all of the points we've identified. And there is our function Y equals negative four thirds cosine of pi X divided by two minus pi divided by three. Well done. We'll catch you on the next one.

Graph each function over a one-period interval.
y = -½ cos (πx - π)

Key Concepts

Period of a Trigonometric Function

Phase Shift in Trigonometric Functions

Amplitude and Reflection

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