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 = 3 sin(2x − π)

Accepted Answer

Hello, everyone. We are asked to identify the amplitude phase shift and period of the given sign function. And then we will sketch its graph considering only one period, the function we are given is Y equals six multiplied by the sign of four X minus pi we are given a coordinate plane to graph. On first recall that the general format of a sine function is Y equals a multiplied by the sign of B X minus C. So when we compare that to our function, Y equals six multiplied by the sign of four X minus pi this will help us match up our values. So we can find our amplitude phase shift and period. Beginning with the amplitude, the amplitude will tell us how far up or and down our Y axis. This sine wave goes recall that traditionally the amplitude amplitude is one here. We can find the amplitude is the absolute value of A A is six. So the absolute value of six is six. So this means our sine wave will go up to six and down to negative six. When we graph it next, the phase shift B shift is found by doing the value of C divided by the value of B. So here C is pi because they are both minus, so it's minus C in the format and it's minus C in our function. So C is just pi B is four, the value directly in front of X. So our phase shift is a positive pi divided by four. The period of the function can be found by doing two pi divided by B. So in our example, B is four, so two pi divided by four will reduce to pi divided by two. So the period here is pi divided by two. So now to find our X and Y values to be able to graph this first recall that for a sine wave, you want four equal sections. So I'm gonna take our period and divide it by four to find the width of each section. So pi divided by two, divided by four is the same as saying pi divided by two, multiplied by the reciprocal of four, which is 1/4. So each section of our graph will be pi divided by eight with. So now thinking about my X and Y values to be able to graph this, I'm gonna write all of my Y values first because I know that my sine wave starts at Y equals zero, then it increases until we hit the amplitude. So that'll be six and then it decreases back down to Y being zero. And then we dip under the X axis to the negative version of our amplitude, which is negative six and then it returns to the X axis where Y equals zero. So now I wanna find the X values to go with that. So to find the first X value, I again recall that typically this starts at zero. So this will be X sub one. However, we have a phase shift. So we need to add the phase shift of pi divided by four. So our first X value is pi divided by four. To find our second X value, we do pi divided by four plus the width that we found. So pi divided by eight. And when I add those together, I get three pie divided by eight. So to find the third X value, I take three pi divided by eight and I add another pi divided by eight for our width. This will give me four pie divided by eight, which is the same as pie divided by two for my fourth X value. I'm gonna put it in here as four pi divided by eight and then add pi divided by eight. So this will give me five pie divided by eight. And last X value five pi divided by eight plus pi divided by eight is six pie divided by eight, which can be reduced to three pie divided by four. Now I have taken into account my amplitude, my phase shift and my period I think we're ready to graph this. So when X is pi divided by four, Y is zero, when X is three pi divided by eight, Y is six. So that is here when X is pi divided by two Y is zero, when X is five pi divided by eight, why is negative six? And when X is three pi divided by four Y is zero. So I have all my points on there. I'm going to try to make a wave without making it too messy. And there we are, this is our sine wave. When we have an amplitude of six, a phase shift of pi divided by four and a period of pi divided by two 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 = 3 sin(2x − π)

Key Concepts

Amplitude of a Trigonometric Function

Period of a Sine Function

Phase Shift of a Trigonometric Function

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