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 = cos(x − π/2)

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 the cosine of in parentheses X minus pi divided by eight. We are given a coordinate plane with an X axis that appears to have pi divided by two intervals and A Y axis that has intervals of one. First recall that a typical cosine function would be Y equals a multiplied by the cosine of in parentheses. BX minus C comparing this to our function, we have Y equals the cosine of in parentheses X minus pi divided by eight. So this will help help us match up our values. So beginning with the amplitude, we can find amplitude by doing the absolute value of the A A. Here, we don't have an obvious number which means it's one. So the absolute value of one is one. So our amplitude isn't actually going to change from what it would be. If it was a standard cosine function looking for the phase shift base shift can be found by doing the C value divided by the B value C for us is pi divided by eight dividing that by B which is one, our fee shift is pi divided by eight. So this means instead of starting at an XX value of zero, we're gonna start an X value of pi divided by eight. And then to find the period of this graph, we do two pi divided by B B here is one. So two pi divided by one is two pi summer period is two pi before I go to make my X and Y table to find the values, I'm going to plot, I'm gonna try to find how far each X value is away from the next X value. So the intervals of this and we can do that by doing the period divided by four. So two pi divided by four that reduces the pi divided by two. So each X value will be pi divided by two away from the previous X value. So making an XY table, my first X comes from the standard positioning X would have been zero, but I want to add the phase shift of pi divided by eight. So our first X value is pi divided by eight. Since our amplitude is one, we don't need to change our values from what they are in a standard cosine function. So our first Y value will be the value one to find our next X value. We do pi divided by eight plus pi divided by two which when you make a common denominator and add, you'll get five pi divided by eight. And the Y value that goes with that is zero. For our third X value, we have five pi divided by eight plus pi divided by two. And again, we need to make this into the common denominator. And when we do, we get nine pi divided by eight. And since our amplitude didn't change, this is going to be negative one for A Y value are fourth X value nine pi divided by eight plus pi divided by two. Again, create a common denominator and we'll get 13 pi divided by eight. And that will have a Y value of zero. And for our final X value, we'll do 13 pi divided by eight plus pi divided by two. Again, creating a common denominator of eight. We will have 17 pi divided by eight as our last X value. And the Y value that goes with that is one. So now that I have the values I'm going to put them on my graph. So we start at pi divided by eight and one, then we drop down to P five pi divided by eight. And that's at zero nine pi divided by eight. The Y value is negative one, 13 pi divided by eight goes back up to A Y value of zero and 17 pi divided by eight brings us up to a Y value of one. So I'm gonna make a wave of this the best I can. And we have a valley as our graph that goes between quadrants one and four. Have a nice day.

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

Key Concepts

Amplitude

Period

Phase Shift

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