In Exercises 43–52, determine the amplitude, period, and phase sh...

Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 1/2 cos (3x + π/2)

Accepted Answer

Find the amplitude phase shift and period of the given function. And sketches graph for one period where we have Y equals 1/ cosine of two X plus pi divided by three. Now we have the general form of a cosine function which is a cosine BX minus C. And this function are amplitude is the absolute value of hate. Our period is given by two pi divided by B and our phase shift is given by C divided by B. We can then see our amplitude A will be 1/7 because it is multiplying our cosine our period will be two pi divided by our B which is two making our period. Just pi finally, we have our face shift. Our phase shift will be negative pi divided by three all divided by two which gives us negative pi divided by six. No, our graph will then start at our phase shift for this period, meaning we will start our first point at native pi divided by six. Now we need five points here for it to work and we can divide our period by four to get all five of our points. So I divided by four will be our increment. In this case. No, since we're starting at negative pi divided by six, our correspondence Y is our amplitude for a cosine graph. Meaning of corresponding Y value is 1/7. Since this is a positive function, we will start the positive amplitude for our Y our next point, we use our increment, we will increment our X by pi divided by four to get our next location giving us pi divided by 12. Now the second point in the cosine graph unless it's vertically shifted will have a Y of zero. We now do the same for our next X value. We increment again by pi divided by four pi divided by three. And this time, our amplitude will be negative. Since we started with a positive amplitude, our middle point will have a negative amplitude. Now we have our fourth point which we increment by pi over four from pi over three. Once again, you get seven pi divided by 12. And this once again is at zero as long as it's not vertically shifted. And finally, we have our fifth point which occurs as seven pi by the by 12 plus pi di by the by four. That is five pi divided by six. And this goes back to a positive amplitude, 1/7 we can now graph our function. We started at negative five or six and one sevens. There we go. It's a pi divided by 12 pi divided by three negative one sevens seven pi divided by 12 and zero. And finally five pi divided by 6, 1/7 we can draw a smooth curve between our points and we now have the graph of cosine along with our amplitude period and phase shift. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

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In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 cos (3x + π/2)