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 2πx

Accepted Answer

Hello, everyone. We are asked to identify the amplitude and period of the given sign function. And then we will sketch it by considering only one period. The function we are given is Y equals negative five multiplied by the sign of four pi X, we are given a coordinate plane to work off of. So first, I'm gonna write down what my general format is for a sine function. And that is that Y equals a multiplied by the sign of B X minus C in parentheses. So for comparison purposes, I'm gonna put ours directly under there. So Y equals negative five multiplied by the sign of four P X. All right, we will start with the amplitude. Amplitude is gonna be how high up on the Y it goes or how low the wave goes. We can find the amplitude by taking the absolute value of A in this example, A is negative five. So the absolute value of negative five is five. So this means that our highest point will be five on the Y axis and our lowest will go down to negative five while we're looking at A I notice A is negative. So this means that our sine wave is going to decrease first and then increase. So typically a sine function starts at 00 and increases, we're gonna start at 00 and decrease next. I need to find the period. So how long does this go on? And so that's two pi divided by B so B is the value directly in front of X. So we have two pi divided by four pi and when we reduce, this ends up being one half, 10.5. Not every day, you get a period that doesn't have a pie in it. All right. So we know the period, we know the amplitude. Let's think about graphing it. First, I recall that sine waves have four distinct sections. So I'm gonna take my period and divide it by four. So I think I'm gonna put it back as a fraction. So one half divided by four is the same as saying, one half multiplied by the reciprocal of four, which is 1/4. So each X value should be 1/8 away from the previous X value. So this is gonna be the width of my X values if you will. And in case it's easier to look at it as a decimal 1/8 as a decimal is 0.125, it might be easier to work with. I haven't decided yet. So next, I recall that when I have a sign function and there is no phase shift, that my first point will be the value zero for X and zero for Y. So next I know that A was negative. So I'm gonna use this information and fill in all of my Y values. So I know A is negative and that the amplitude is five. So traditionally, my second point if A was negative would be negative one, but since our amplitude is five, it would be negative five, Next, it goes back to zero. After that, typically, it would increase to a positive one. But since my amplitude is five, it'll increase to positive five and then back to zero. So I have my Y values, I need to find my X values. And I think I'm gonna make my X values fractions, not decimals just to help myself. So if my first X value is zero, my second X value will be zero plus that width we found which was 1/8. So the second X value is 1/ to find our third X value, we take our second X value of 1/8 and we add another 1/ which is 2/8 which reduces to 1/4. So our third X value is 1/4 to find our fourth X value. We're gonna take our 1/4 which I'm gonna rewrite as 2/8 and add 1/ to it. So this will give me 3/ and to find our last X value, it should be equal to our period of one half. But let's verify that. So 3/8 plus 1/8 is 4/8 which is in fact one half. Again, if you prefer to look at these as decimals, please go right ahead. I just think I'm gonna work a little easier with fractions right now. Looking at the scale of my Y axis, I notice it's in ones. Meanwhile, the scale of my X axis is in quarters or 0. fives. So my first point is the 0.0.0 right on the origin, next 1/8 and negative five. So if my scales in 1/4 it means I'm about halfway between the first two lines and then I'm going down to negative five. Then for 1/4 we're back up at zero. So 1/4 is that first line after the Y axis on the right 3/8 will fall between 1/4 and one half about halfway between and then go up to five on the Y axis and then one half is zero. So one half should be our second line over to the right and back on the zero. So now I will try to connect these smoothly in a wave and though it is not exactly perfect, it will connect the dots and this is what our graph looks like. Have a nice day.

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