In Exercises 75–78, graph one period of each function.

y = −|3 s...

Question

In Exercises 75–78, graph one period of each function.

y = −|3 sin πx|

Accepted Answer

Hello, everyone. We are asked to draw the graph of the function for one period. The function we are given is Y equals the negative absolute value of 17 sine of four pi X. We have a coordinate plane. Our X axis is labeled in 0.2 increments. Our Y axis is labeled in one unit in increments. To begin with, we are going to treat Y equals negative absolute value of 17 s of four pi X as if it's the negative 17 multiplied by the absolute value of the sign of four pi X, we can do this through different rules of sign. And then we're gonna work with negative 17 s of four pi X without the absolute value symbol to get both the amplitude and period. And then we'll reapply the absolute value at the end. So to begin with, we need to find the amplitude to know how high or low this goes. So the amplitude is the absolute value of A in our case A is negative 17. So the absolute value of negative 17 is 17. So instead of having a range between negative one and one, it will now go from negative 17 to 17. Next, we need to find the period. The period is two pi divided by BB is the value directly between the words and the variable X. So two pi will be divided by four pi which reduces to one half. So the period of our function is one half. Now, I want to try to find X and Y values and I also recall that my A was negative. So this is going to be flipped upside down from our normal sine wave either way without a phase shift, our first point is gonna be 00. And then to find what spacing we have between our X values, we need to take the period divided by four. So this is gonna be one half divided by four, just the same as one half multiplied by the reciprocal of four, which is 1/4 which is ends up being 1/8. So each X value is 1/8 away from the previous X value. So my next de value after zero will be 1/8. And in a standard sine wave, this would be one, a negative sine wave would be negative one. But because our amplitude is 17, this is negative 17, adding 1/8 to 1/8 we cut 1/4. And again, thinking of my traditional Sine wave, this would be a zero and it still is A Y value of zero, adding 1/8 to 1/4 we get 3/8 is our next X value and this would be a negative one in a standard sine wave, a negative sine wave, it would be a positive one. And with an amplitude of 17, this is a positive 17 next X value adding 1/8 to 3/8 gives us one half. And that puts us back at zero. Here is where we put the absolute value back in. Since this is an absolute value, we will turn all of the positive Y values negative. So we will use the negative of all positive. Why values? So this means our first point of 00 stays the same. Our second point 1/8 and negative 17 stays the same because it is a negative. We don't need to change it. Next. Our point 1/4 0 stays the same. The point 3/8 and a positive 17 will now be 3/8 and negative 17. And finally, the 0.1 half zero remains one half zero. So now we'll sketch this on our graph. Our increments are not precisely 1/8 apart. So we're gonna estimate a little bit starting with 00. That is our origin and then 1/8 negative 17, 1/8 is about 0.125. So if this is 0.1 I'm gonna go over a little bit more, go down to where negative 17 should be 1/4 0. So 1/4 is 0.25. So it'll be halfway between 0.2 and 0.3 and on the X axis 3/8 is 0.375. So a little bit before 0.4 oops and then we need to go down to negative 17. Here we go. And then one half, which is the same as 0.5 and zero. So that is right on the line. And now we need to make a wave out of this. So it's kind of like two parabolas next to each other. But this is the negative absolute value of 17 sine of four pi X have a nice day.

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In Exercises 75–78, graph one period of each function. y = −|3 sin πx|