In Exercises 7–16, determine the amplitude and period of each fun...

Question

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function.y = 3 sin 1/2 x

Accepted Answer

Hello, everyone. We are asked to identify the amplitude and period of the given sign function. And then we will graph it considering only one period. The function we are given is Y equals five multiplied by the sign of 1/4 X. We are given a coordinate plan for our sketch. First recall that the general format for a sine function is that Y equals a multiplied by the sign of in parentheses B X minus C. When we compare this to our function, Y equals five sign of 1/4 X, we notice we have no C so we won't have any sort of phase shift to deal with. First, we're gonna find the amplitude. The amplitude is basically like saying that our normal sine wave goes up to one and down to negative one. Will this change? Will it be greater? Will it be smaller? So our amplitude is the absolute value of A A is the value directly in front of the word sign. And in this case is five. So the absolute value of five is five. So our amplitude is five. So instead of going up to one, it'll go up to five instead of going down to negative one, it'll go down to negative five. Next, we're gonna find the period of this sine function. Recall, we find the period by doing two pi divided by B. So in our case B is 1/4 so we have two pi divided by 1/4 and I can rewrite that as two pi multiplied by the reciprocal of 1/4 which is four. So our period is going to be eight pie. Now that we've found the amplitude in the period, I'm gonna start finding the things I need to be able to graph this. I recall that sine functions have four distinct sections increasing, decreasing, decreasing, increasing. If you think about the wave shape. So to find the distance between each X value, I'm going to do the period divided by four. So eight pi divided by four is chia. So this means between every X value, there will be two pi. So I'm gonna make an X Y table for my points and since there is no phase shift, the first point will be from there, I'm gonna fill in the rest of my Y values before going back to my X values because I know there's a pattern to the Y values. So we start at zero and then it increases to the amplitude of five and then I'll decrease back down to zero. So I would be zero and then it decreases to negative five and increases again, for Y to be zero. So I have all my Y values, I need to find the X's that go with it. So to find my second X, I've labeled X sub two, I take my first X which is zero and I add that width or the space between the XS I found out by doing the period divided by four. So two pi so my second X value is to find my third X value. I take the second X value. So two pi and I add our width which was also two pi and that gives me four pi. So my next X value is four pi. So my fourth X value will be the third one. So four pi plus the width of two pi. So this will be six pi and my final X value if we're only graphing for one period should be equal to the period we found. So let's check six pie plus two pie does equal eight pie, which was our period. So I know that something went right. All right. Now, I'm gonna put these points on my graph. It is very nice that the X axis is labeled in pi because it makes it a lot easier to graph. So we're gonna start at 00. And then when X is two pi Y is five, so two pi and then up to five, then for X is four pi Y is zero. When X is six pi why is negative five. And when X is eight pi Y is zero again. So I have all of my points. I'm gonna try to make as nice of a curve as I can. It might not be perfect. All right, I will never draw a perfect sine wave, but that is pretty close for me. So we have all of our points. Our amplitude is five, our period is eight pi and this looks like our sine wave. Have a nice day.

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function.y = 3 sin 1/2 x

Key Concepts

Amplitude

Period

Graphing Trigonometric Functions

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