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 cos (x + π)

Accepted Answer

Hello, everyone. We are asked to find the amplitude phase shift and period of the given function and sketch its graph for one period. The function we are given is Y equals negative five multiplied by the cosine of in parentheses, X plus three pi we are given a coordinate plane to work with where our X axis is in pi divided by two increments. And our Y axis is in one increments to begin with. I recall that the general setup of a cosine function is that Y equals a multiplied by the cosine of in parentheses BX minus C. So comparing that to our function, we have Y equals negative five multiplied by the cosine of X plus three pi in parentheses. So starting with our amplitude amplitude is found from the absolute value of A. So in this example, A is negative five. So amplitude is the absolute value of negative five which is five. This means instead of going up to one, it will go up to five, instead of going down to negative one, it'll go down to negative five on the Y axis. Next, I want to find the phase shift. So typically we would start at the 0.01. This will tell us where we're starting and we can find the phase shift by doing the value of C divided by the value of B. So matching up with our functions, we notice in our base function, it is A minus C and in ours, it's a plus three pi, this means RC is actually gonna be negative three pi and B is the value in front of X since we can't see it, it's an implied one. So our fee shift is gonna be negative three pi and then to find the period, we would take two pi and divide it by B. So two pi divided by B is one, our period here is still two pi. So now we'll work on finding the points we are going to put on our graph. So I'm gonna make an XY chart and I'm actually gonna start with what we get if we just have the cosine of X equals Y. So what our basic one typically looks like and that's when we have zero for XY is one, when we have pi divided by two, for X Y is zero and we have pi for XY is negative one when we have three pi divided by two for XY is zero. And when we have two pi for Xy is one. So I started with the normal one, the one without any phase shifts or amplitude changes to help me find the values for our function. So ours again is Y equals negative five, multiplied by the cosine of X plus three pi in parentheses. So I'm making my XY table for this. So our first X value will be our normal X value from when we had no phase shift plus the phase shift. So our first X is zero plus negative three pi So this means our first point here is that negative three pi and we're gonna multiply Y by our A value, our A value is negative five. So this is gonna start at negative five. So our first point is negative three pi negative five because the period didn't change, we will have the same increments. So each X value is pi divided by two spaces apart. So X is pi divided by two away from the next X. So if I add pi divided by two to negative three pi, I get negative five pi divided by two. And we're gonna again think about our original Y value if we had no phase shift or amplitude change. And it was zero. Multiplying that by the a of negative five, it is still zero, adding another pi divided by two to my X value. I get negative four pi divided by two, which is the same as negative two pi and then our third X value without any changes was negative one multiplying that by R A, this will be a positive five, adding another pi divided by two. We'll have negative three pi divided by two is our X value zero multiplied by negative five is zero. And our final X value will add another pi divided by two to this which will give us negative two pi divided by two, which is the same as negative one pie. And the Y value of one multiplied by negative five will give us negative five. So now I have points that I can put on my graph. So I have negative three pi negative five, negative five pi divided by 20, negative two pi and five negative three pi divided by two and zero and negative pi and negative five. So trying my best to connect those in a wave. Our graph looks a little bit like a mountain in quadrants. Two and three 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 cos (x + π)

Key Concepts

Amplitude

Period

Phase Shift

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