In Exercises 7–11, determine the amplitude, period, and phase shi...

Question

In Exercises 7–11, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = −3 sin(π/3 x − 3π)

Accepted Answer

Hello, everyone. We are asked to find the amplitude phase shift and period for the given function and sketch its graph for one period. The function we are given is Y equals negative seven, multiplied by the sign of open parentheses. Pi divided by four, X minus four pi closed parentheses. We're given a coordinate plane where the X axis begins at the value 15 and each is counted by one unit and the Y axis is in a scale of one and it begins at zero and goes up to positive V and down to negative V. First recall that the general format of a sine function is that Y equals a multiplied by the sign of open parentheses. BX minus C closed parentheses. So comparing that to what we have, we have Y equals negative seven multiplied by the sign of pi divided by four X minus four pi. So this means A is gonna be negative seven. Well, we're gonna use that for our amplitude. So recall that we can find amplitude by finding the absolute value of A. In our case, we just said A is negative seven. The absolute value of negative seven is seven So this means our sine wave will go up to positive seven and down to negative seven. I also like to make a little bit of a note to myself here that since A is negative, we are going to start at zero and decrease because we are going the opposite direction of our traditional sine wave. Next, we want to find the phase shift. So typically we start at the 0. but we might not depending on what values we're given. So the phase shift can be found by doing the value of C divided by the value of B. So in this case, C is four pi we're gonna divide that by pi divided by four, which is the same as saying four pi multiplied by the reciprocal of pi divided by four which is four divided by pi. And when I cross reduce the pies, we get that our phase shift is 16. So instead of starting at the 0.00 this phase shift means we're gonna start at the 0.16 0. So, so far we have our phase shift, we have our amplitude, we need to find the period. So is it gonna be stretched out or is it gonna be squished down? So to find the period, we do two pi divided by B. So two pi divided by B in our case is pi divided by four. So this is the same as saying two pi multiplied by the reciprocal of pi divided by four, which is four divided by pi cross reducing the pies because that our period here is eight. OK. So now that we know this, let's start making our graph. Well, I'm gonna make a co table for my X and Y values just to help me keep track. So because our period is eight, we want to find out how far away are each of these values from each other? Well, we know it's gonna start at 16 now, right? We said that phase shift means we are starting at the 0.16 0. So how far is it before I choose another one? Well, to find out the increments or my intervals, I'm gonna take the period divided by four. So eight divided by four is two. So this means my next X value is gonna be the value 18. And in our traditional sine wave, we would have gone from up to one on the Y axis. But since this is a negative and our amplitude is seven, we are going to decrease. So our Y value will be negative seven and then it goes back to the X axis with zero. So what our X value is is 18 plus two. So 23 of the 0.20 0. And then my next X value would be 22. And this is when our sine wave is gonna go up to the positive seven and then it will return to our X axis at 24. So 24 0, again, I know this because typically we would have 00 and then it would be pi divided by two up to one. And then pie would be at zero three pi divided by two would have been a negative one and two pi would have been back at zero. So I followed the same pattern for my Ys but I multiplied them by my amplitude. So let's plot this. So 16 0 will be on the X axis 18, negative seven will be right about here. 20 0 will be back on the Y axis. I'm sorry. The X axis 22 7 means we go 22 on the X seven on the Y and 24 0. And I'm gonna connect my points with a wave and this is one period of the function Y equals negative seven sign of open parentheses. Pi divided by four X minus four pi closed parentheses have a nice day.

In Exercises 7–11, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 sin(π/3 x − 3π)

Key Concepts

Amplitude

Period

Phase Shift

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