In Exercises 7–16, determine the amplitude and period of each fun...

Question

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function.

y = -sin 2/3 x

Accepted Answer

Hello, everyone. We are asked to identify the amplitude and period of the given sign function. Then we are going to sketch the graph by considering only one period. The function we are given is Y equals the negative sign of 2/7 X. We are given a coordinate plane to sketch. On first recall that the general format of a sine function is that Y equals a multiplied by the sign of B X minus C in parentheses. For example, we have Y equals negative sign of 2/7 X. So we do not have AC looking for the amplitude amplitude is gonna be the absolute value of A. In this case, A is negative one. So the absolute value of negative one is one. So this means that the highest point of our sine wave will be at one and the lowest point will be at negative one while I'm looking at A, I notice it is negative. So I know that my wave is gonna start by decreasing. So A is negative. So the sine wave will start decreasing. And this is gonna come in handy when we start graphing. Next, we want to find the period we can find the period of a sine wave by doing two pi divided by B. So we have two pi divided by B in our case is 2/7. So knowing that this is a complex fraction, I can do two pi multiplied by the reciprocal of 2/7 which is seven halves. And when I cross reduce, I get that my period will be seven pi. So I've identified the amplitude of one and the period of seven pi. So now let's get things going. So we can graph recall that in a sign function, we have four separate sections and we want to divide our period into four equal sections. So we're gonna take the period and divided by four. So seven pi divided by four is gonna be the wi of each section. So if I were to graph this, I'm gonna start by making my X Y chart for my points. I recall a sine wave starts at the 0.0. So that will not change here from there. I want to find my next X value which I'll call X two. So I'm gonna take zero and add seven pi divided by four to it, which will be seven pi divided by four. Recall that we said that A is negative. So our sine wave is going to decrease first. So this means the Y value that goes with seven pi divided by four is going to be the negative of our amplitude So negative one. Next, the section where our sine wave will increase back to Y equaling zero. So I'm going to put zero is my next Y value. I need to find the value for X and we'll do seven pi divided by four plus another seven pi divided by four. And this will give me 14 pi divided by four which can be reduced the seven pi divided by two. So that's my third X value. Seven pi divided by two. My next X value is gonna be when the sine wave increases to the amplitude of one. So to find that fourth value, I'm gonna take that 14 pi divided by four that we had before we reduced and add seven pi divided by four to that. And I will get 21 pi divided by four. So that is my fourth X value. And for my final X value where Y goes back to being zero, this should equal seven pi but I'm gonna do the math just to double check myself. So I have 21 pi divided by four plus seven pi divided by four, which gives me 28 pi divided by four, which does equal seven pi. So I know that I've done my math correctly because our last X value should be what the period is. So seven pi, all right now that we have all of our values, we're gonna sketch it onto our graph. We'll start with 00. Next we need to find seven pi divided by four. Um That is greater than one pie but less than two pie. So it'll be in between pie and two pie and down at negative one for the Y value. Next seven pi divided by two. That's gonna be a little bit more than three pie, but less than four pie. And that goes back to A Y value of zero, next 21 pi divided by four. So this is a little greater than five pie, but less than six pie. And that's gonna go to a Y value of one. So I said greater than five pie, but less than 65. So roughly here and then seven pie will be back at zero. So our sine wave should look generally like this. It starts at 00, it decreases to negative one, back to zero up to one and then back down to zero. Have a nice day.

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = -sin 2/3 x

Key Concepts

Amplitude

Period

Graphing Trigonometric Functions

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