In Exercises 1–6, determine the amplitude and period of each func...

Question

In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function.

y = 1/2 sin π/3 x

Accepted Answer

Hello, everyone. We are asked to find the amplitude and period of the given function and sketch its graph for one period. The function we are given is Y equals one third multiplied by the sign of pi divided by six X. We are given a coordinate plane where the X axis is in increments of one and the Y axis is in increments of 0.1 to begin with. I recall that a sine function is set up as Y equals a multiplied by the sign of open parentheses. BX minus C matching that to what we have, we have Y equals one third multiplied by the sign of pi divided by six X. So this means in our case A is one third, B is pi divided by six and C would be zero starting with the amplitude amplitude is how high or low the graph will go and it is the absolute value of A. So we'd have the absolute value of one third, which is one third. So our amplitude is one third. So instead of going all the way up to one and all the way down to negative one, we will go up to one third and down to negative one third. Next, it rest for the period. The period can be found by two pi divided by B. So this would be two pi divided by pi divided by six, which is the same as two pi multiplied by the reciprocal of pi divided by six which is six divided by pi. So when I cross reduce, I can cancel the pies and I get that the period is 12. So before I start sketching this, I recall that our standard sine wave has X and Y values that start at 00. And then our pi divided by two for X one for Y pi for X zero for Y three pi divided by two for X negative one for Y and two pi for X zero for Y. So now taking into consideration that our period is 12, we need to find the intervals or how far apart each wave will go. So I'm gonna say to find an interval, I'm going to do the period divided by four. So 12 divided by four is three. So each of our X values will be three units apart. Since there is no phase shift, we will still start at the 0.00. Our next X value will be zero plus three which is three. And we're gonna take our Y value and multiply it by the amplitude. So one multiplied by one third is one third. Next we're gonna add three to our X value and get six zero, multiplied by the amplitude of one third is zero. Adding another nine to the X value. We get, I'm sorry, adding another three to the X value, we get nine and then the negative one will be multiplied by the amplitude of one third. So that'd be negative one third. And finally adding another three, we have an X value of 12 and a Y value of zero. So these are our points and we're gonna graph them as close as we can. Um One third is 0.3 repeating. So I'm gonna just put it on the 0.3 line. So starting with 00 is right here at the origin and then three on the X one third on the Y. So roughly three lines up on the Y axis six for the X zero for the Y. So we're back on the X axis nine and then we go down to negative one third and then 12 goes back up to zero. So the watt on the X axis, I'm gonna connect my dots in a wave to the best of my ability. And this is what the sketch of Y equals one third sign of pi divided by six X looks like. Have a nice day.

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In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function. y = 1/2 sin π/3 x