In Exercises 17–30, determine the amplitude, period, and phase...

Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2x + π/2)

Accepted Answer

Hello, everyone. We are asked to identify the amplitude phase shift and period of the sine function then sketch its graph by considering only one period. The function we are given is Y equals negative three multiplied by the sign of in parentheses, two X plus pi divided by four. We are given a coordinate plane where the X axis is labeled in radiant and the Y axis is labeled um with a scale of one. First recall that the general format for a sine function is that Y equals a multiplied by the sign of in parentheses B X minus C. So if we compare this to the function, we are given where Y equals negative three multiplied by the sign of two X plus pi divided by four, this will help us find our amplitude phase shift and period starting with the amplitude, the amplitude or how high this goes on the X axis or low comes from the absolute value of A. So R A is negative three. So the absolute value of negative three is three. So this means instead of going up to one, it's going to go up to three instead of going down to negative one, it's gonna go down to negative three. Also note that it is negative. So instead of starting at zero in increasing, we're gonna start at zero for the X and decrease. So we're gonna just note that this starts decreasing because it is a negative A. So it starts decreasing. Next, the phase shift. So this is how far off of our traditional sine wave we're gonna go left or right. And we can find this by doing the value of C divided by B looking quickly, I thought C was pi divided by four. But then I noticed that the signs are different between the general function and our function in the general function. It's a negative C or minus C. Here we have a positive pi divided by four. So that means it actually must be a negative pi divided by four because the negative minus itself would have given us the positive. So, and then we divide by B which is two. So this is the same as negative pi divided by four, multiplied by the reciprocal of two, which is one half. So our phase shift is negative pi divided by eight. And lastly here we need to find the period or how long is this? We can find the period by doing two pie divided by B B in our case is two. So two pi divided by two. This means the period here is pie. So we almost have all the information we need, we do need to find some points before we can graph this. So first, I know I want to have four equal sections. So I'm gonna divide our period by four, we get our four widths to know how wide each section needs to be. So pi divided by four, we're gonna treat as pi divided by four. So typically we would start with X as zero. However, with our phase shift, we are gonna shift negative pi divided by eight to the left. So instead of zero, we will have negative pi divided by eight as our first value. And that's gonna be when Y is zero to find the next section, we're gonna do negative pi divided by eight plus pi divided by four. And when I add those together, I get pi divided by eight. So that is my next X value pi divided by eight. And as I said, typically, our sine wave would increase. But since R A was negative, it's going to decrease. So pi divided by eight will have a Y value of negative three. Next, I want to find the next value. So I do pi divided by eight plus pi divided by four. And when I combine those, I get three pie divided by eight. So that is my next X value. And this would be uh the sine wave increasing back to the X axis. So zero is our Y value to find the next one. We do three pi divided by eight plus pi divided by four. And this will give us five pie divided by eight. So that's our next X value and that's gonna be when we increase to three. And our last value to make this be a full period of pie, we would do five pi divided by eight plus pie divided by four, just seven pi divided by eight. So seven pi divided by eight is my last X value and that'll bring us back to the X axis. So Y equals zero. So now let's get these points on the graph. Negative pi divided by eight on the X axis would be between zero and negative pi divided by four. And since the Y value is zero, it will be right here on our X axis. Next pi divided by eight and our Y value is negative three. So X being pi divided by eight would be halfway between zero and pi divided by four. And then we go all the way down to negative three. Next three pi divided by eight which would be in the middle of pi divided by four and pi divided by two. And that goes back to the X axis. So Y is zero five. Pi divided by eight is between pi divided by two and three. Pi divided by four and has a Y value of three. And then seven pi divided by eight is between three pi divided by four and pi and that gets back to zero on the X axis. So Y equals zero now that we have all of our points plotted, let's graph this curve. So it's a wave. So it's got a little bit of a loop to it. But we started at negative pi divided by eight went down until we got to Y equals negative three. Increased back to the X axis went up to Y equals three and decreased down to the X axis again. And this is our graph and have a nice day.

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2x + π/2)

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Phase Shift of a Trigonometric Function

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