In Exercises 17–30, determine the amplitude, period, and phase sh...

Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = 1/2 sin(x + π/2)

Accepted Answer

Hello, everyone. We are asked to identify the amplitude phase shift and period of the given sign function. Then we're gonna sketch its graph by considering only one period. The function we are given is Y equals 1/ multiplied by the sign of X plus pi divided by four. We are given a coordinate plane to graph our sine wave. On recall that the general format for a sign function is that Y equals a multiplied by the sign of B X minus C. So matching that to our function, we have Y equals 1/ multiplied by the sign of X plus pi divided by four. So let's use this information to graph and to find our amplitude phase shift and period starting with the amplitude, which is how high the graph goes on the Y axis or how low it goes here. Our standard amplitude is one, we know that R A is 1/4 because A is the value directly in front of sine. And we could find the amplitude by doing the absolute value of A. So the absolute value of 1/4 is 1/4. So our amplitude is 1/4. This means instead of going up to one on the Y axis, it will only go up to 1/4 instead of going down to negative one, it will go down to negative 1/4 from there, we have a P shift. So phase shift can be found by doing the value of C divided by the value of B looking quickly matching up our function to the base function. I would think that pi divided by four is C but notice the signs in front of them pi divided by four is positive where in the base formula, it's a minus C or a negative C. So this means C must actually be negative pi divided by four. And when we divide that by B B is the value directly in front of X and that's one. So our P shift is negative pi divided by four. Next finding the period of our sine wave, the period can be found by doing two pi divided by B. So two pi divided by R B was one. So the period here is two pi so we have a lot of our information. So we're gonna start trying to graph this. The first thing I'm going to do is I'm gonna break our period into four equal sections. So I wanna find out what is the width of each section? So I'm gonna do the period divided by four. So two pi divided by four would reduce two pi divided by two. So this means each of our sections will be a width of pi divided by two to find out our starting point. Usually a sine wave starts at zero, but we need to subtract our phase shift or add our phase shift. Since this is negative, we will be subtracting. So zero minus pi divided by four means our first X value will be negative pi divided by four. So I'm gonna make an X Y table and I'm gonna actually shorten that line. So I have a little bit more space to work. And I know my first X value. Once I take into account, my phase shift is negative pi divided by four. And that's gonna be when Y is zero because our sine wave will start at the X value or sorry, the Y value of zero, my next X value will come from adding the width we just found. So pi divided by two to our starting point. So negative pi divided by four plus pi divided by two would give us pi divided by four. So that's my next X value and our sine wave increases in the first interval. So this will increase to our amplitude of 1/4. Next, I'm going to add pi divided by two, two pi divided by four to get our next section. And when I do so, I'll have three pie divided by four. And over the next interval, our sine wave will decrease back to the X axis. So Y will be zero next to find the next X value I have three pi divided by four plus pi divided by two. This will give me five pi divided by four. And in this interval, the sine wave will decrease to the negative of our amplitude. So negative 1/ and our final interval increases back up to where Y equals zero. And the X value that will go with that we will find by doing five pi divided by four plus pi divided by two, which is seven pie divided by four. So we have all of our values. Now let's put them on a graph. So again, recall our sine wave starts usually at the origin. But since we have a phase shift and the phase shift is negative pi divided by four, this will start at negative pi divided by 40, then we increase to pi divided by 4, 1/4. So our Y axis is labeled in decimals. Recall that 1/4 is the same as 0.25. So this will be here and then three pi divided by four brings us back to the X axis five pi divided by four will be negative 1/4 on the Y. So negative 40.25 and then seven pi divided by four brings us back up to zero on the Y. So on the X axis and now when I connect those, we have a wave that looks something like this, it increases and then it decreases back to the x axis dips under the x axis and returns to it. 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 = 1/2 sin(x + π/2)

Key Concepts

Amplitude

Period

Phase Shift

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