Graph each function over a two-period interval.
y = -2 +...

Question

Graph each function over a two-period interval.

y = -2 + (1/2) sin 3x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function consider two periods. Our function is Y equals negative four plus one third sine of four X. We're given a blank graph. We have a vertical Y axis and a horizontal X axis. They come together at the origin. The domain for what's drawn for our X axis is from 0 to 4 pi divided by three. And that's marked off in increments of pi divided by three. The range for what's drawn for our Y axis is from negative five to positive one and that is marked off in increments of one. All right. So our standard formula for a sine function is we recall from previous lessons, Y equals C plus a multiplied by the S of and then all in parentheses. B multiplied by the quantity of X minus D, close both sets of parentheses. So we can compare our given function with the standard formula and determine our ABC and D and that can help us graph our function. So comparing these, we see the C value, what's being added to our sine function is negative four and the C value has to do with a vertical shift because our C value is negative for we have a vertical shift down four units A is what's being multiplied by our sine function. Here A is one third. A has to do with the amplitude. The amplitude is the absolute value of A and here because A equals one third, it's the absolute value of one third which is one third. So our amplitude is one third, B is what's being multiplied by our X inside of that sine function. Here, we can see that B is four, B has to do with the period. The period of a sine function we recall is two pi divided by B. So here we'll have two pi divided by B is four which simplifies to be pi divided by two. So our period is pi divided by two because we're considering two periods will go all the way from zero to pi. Now D is what's being subtracted directly from that X inside of the function. And here nothing is being added or subtracted. So our D value is zero, D has to do with the phase shift. And because D is zero will have a phase shift of nothing, no phase shift. All right. So if you can look at that information and graph it right from that, that's fantastic. But what we can also do at this point is create an XY table, we'll figure out our five key values for X for one period and then we'll repeat another four of them. We plug those in for X in our given function. Come up with the corresponding Y values plot the points connect the points. And then we've got our function graph. So because we have no phase shift, our first key point for the X value will be zero. And then we're going to add 1/4 of our period to that. So if our period is pi divided by two, 1/4 of that will be pi divided by eight. So we'll be adding pi divided by eight to each subsequent X value. So zero plus pi divided by eight is pi divided by eight pi divided by eight plus pi divided by eight is going to simplify to be pi divided by four, add another pie divided by eight and we get three pie divided by eight, add another pi divided by eight and we're halfway there. We at one period pi divided by two, I'll create another table for this second period. So add pi divided by 82 pi divided by two. We'll get five pi divided by eight. Add another pie divided by eight. We're at three pie divided by four, add another pi divided by eight. We're at seven pi divided by eight and add one more and we're at pi there's two periods. So now we'll plug in our X values into our function. I'll show you the first one. So our function was Y equals a negative four plus one third multiplied by sign of, then your calculator will give you parentheses and inside it will be four multiplied by the X value. So here the X value is zero. Plug all that in hit enter and you will get a Y value of negative four. Now if you need to pause the video and plug in for all of these different functions, go for it. If you plug pi divided by eight in for X, you'll get a Y value of negative 11 3rd, which is approximately negative 3.7 which can be a little bit easier to visualize for graphing. If you plug in pi divided by four, you'll get a Y value of negative four. If you plug in three pi divided by eight for X, you'll get a Y value of a negative 13 divided by three which is approximately negative 4.3. If you plug in pi divided by two for X, you'll get a Y value of negative four. If you plug in five pi divided by eight, you'll get a Y value of negative 11 3rd. If you plug in three pi divided by four, you'll get negative four. If you plug in seven pi divided by eight, you'll get negative 13 3rd. And if you plug in P you'll get negative four. Now for me, the hardest part is graphing this because our given graph is in increments of pi divided by three. And our X values are all with a base of two or four or eight. So you need to convert everything to a common denominator of 24 to understand where things go. So for our first point at zero, negative four, that's right along the Y axis, right at zero, negative four, next X value is pi divided by eight which is going to be a little bit past the Y axis. It's a little bit past zero on here. And then that has a Y value of negative 3.7. The next point on here that we're graphing is pi divided by four, negative four. So we're heading down four units and over pi divided by four units which is a little bit before pi divided by three. Next, we've got three pi divided by eight, negative 13 3rd. Three pi divided by eight is just a little bit past pi divided by three. Next is pi divided by two negative four. Pi divided by two is halfway between pi divided by three and two pi divided by three. Next, we've got five pi divided by eight negative 11 3rd. Five pi divided by eight is just before two pi divided by three. Next three pi divided by four negative 43 pi divided by four is a little bit past two pi divided by three to be precise. And next we've got seven pi divided by eight negative 13 3rd, seven pie divided by eight is just a little bit before pie. And then finally, we've got pi negative four and that one's on here, that one's directly on here. So mine's not perfectly evenly space. It's, it's an attempt. But next, you want to smoothly connect these and there we have Y equals negative four plus one third, multiplied by the sine of four X. And there's our assign function well done. We'll catch you on the next one.

Graph each function over a two-period interval.
y = -2 + (1/2) sin 3x

Key Concepts

Sine Function and Its Properties

Amplitude, Frequency, and Vertical Shift

Period of a Trigonometric Function

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