Design a sine function with the given properties.

It has a period of $12$ with a minimum value of $-4$ at $t=0$ and a maximum value of $4$ at $t=6$.

Video transcript

Hey, everyone. In this problem we're asked to create a sine function. Out of the given information, we have that the period is equal to 6, the minimum value is negative 6 when x is equal to 1, The maximum value is 10 when x is equal to 4. We have 4 answer choices, option a through d, each containing a different sine function and we are going to come back to these as we work through the problem. So let's recall a general sine function is going to be written as y is equal to a multiplied by sine of bx plus d, plus d. So in order to write the sine function that represents the information, we need to find all of these values a, b, c, and d. Alright. So we're gonna say that the amplitude is represented by a. Okay. Recall that in our equation, a represents our amplitude. So for a, our amplitude, we want to think about the vertical range. Remember that for a trigonometric function, the amplitude is gonna be that vertical range divided by 2. It's going to be half the vertical range, it's from that midpoint up or down to the maximum or minimum. Now our vertical range here, we can find because we're given the minimum and maximum value. So the distance between the maximum value, which is 10, and the minimum value of negative 6 is going to be 10 minuteus negative 6, and we want to divide that by 2 to get our amplitude. This is going to give us 16 divided by 2, and so our amplitude is going to be 8. Alright. So we found our a value, let's move to the next thing that we need to find, and that's gonna be our b value. Now recall that the b value, that coefficient on the x inside of our sign, is related to the period of the function. And so that b value is going to be equal to 2 pi divided by the period t. Now we're told that the period t is 6, and so in our case B is going to be 2 pi divided by 6, which we can simplify and write as pi divided by 3. So we have a, we have b. Now the next thing we're going to do is we're going to find this vertical shift d, okay, And d, again, that's that vertical shift, so it's gonna represent how much that function is moved either up or down away from the x axis. So for this vertical shift, we want to look at the midpoint between our minimum and maximum values. That's going to be kind of our baseline and that's exactly where that shift is going to lie. Okay, so we're gonna look at the midpoint of the max and min value. Now when we were looking for the total distance between the maximum and minimum, we subtracted those two values. When we're looking for the midpoint, we're going to add them. So we get that the distance d is equal to the maximum 10 plus the minimum of negative 6, and we're gonna divide that by 2. This gives us a value of 4 divided by 2, which is just equal to 2. Okay, so that means that our function has been shifted up 2, and if we put this all together, it makes sense. Okay, if we shift our function up 2, okay, so we're resting at 2, Then we add our amplitude of 10, we get to the or sorry, our amplitude of 8, we get to that maximum value of 10. If we do it again, we start at that baseline, or that shift of 2, we subtract our amplitude of 8, we get to our minimum of negative 6. Alright, so the last thing we want to do is figure out c. There's a couple of different ways to do this. What I'm going to do is just substitute one of these points into our equation. Okay? And the points I'm talking about are these minimum and maximum values. Okay? Those represent points. So what I'm gonna do is I'm gonna take our function, which is now y is equal to 8 multiplied by sine of pi divided by 3x, plus c, plus 2, And I'm going to substitute the point 1, negative 6. When we do that, we get that negative 6 is equal to 8, multiplied by sine of pi divided by 3, plus c, plus 2. We can subtract 2 from both sides. Okay. We want to isolate and solve for c. So we get negative 8 is equal to 8 sine pi divided by 3 plus c. That tells us that sine of pi divided by 3 plus C is going to be equal to negative 1. When does sine equal negative one? Well, sine is going to equal negative one if the angle, pi divided by 3 plus c, is equal to 3 pi divided by 2. This is one solution, but we have to remember that this repeats every 2pi, so we can write this as plus 2pi multiplied by n for n an integer. Alright. So let's go ahead, subtract pi divided by 3 to isolate for c. Okay. We get that c is going to be 3 pi divided by 2, minus 5 thirds, plus 2 pi multiplied by n. This tells us that c is going to be let's find the common denominator. So we have a denominator of 2, a denominator of 3. Our common denominator is going to be 6. So we can write this as 95 divided by 6 k, multiplying the 1st fraction by 3, minus 2 pi divided by 6, multiplying the second fraction by 3 or sorry, by 2. And then we have our plus 2 pi x, and this gives us that c is going to be equal to 7 pi divided by 6 plus 2 pi n. Now, let's remember the values of c that we can have. Remember that for c, we want it to be between negative pi and pi. Okay? 7 pi divided by 6 is bigger than pi, so what we wanna do is we wanna use a negative end value. Okay. We've written that n can be 0, 12, but it can be any integer. We were gonna use that negative end value, we're gonna subtract 2 pi. Okay. So we're using n is equal to negative one here. So what we're doing is finding an angle that is coterminal to 7 pi divided by 6, that falls in that range from negative pi to pi. And so we're subtracting 2 pi, and we get a c value of negative 5 pi divided by 6. Alright, so we started, we found our amplitude, we found the value of b, we found the value of d, we found the value of c. Now we can write our entire function altogether as what? And what we found is that our function is going to be y is equal to 8 sine of pi divided by 3x minus 5 pi divided by 6 plus 2. Okay. And that is the final function representing all of that information we were given. If we go back up to our answer choices, we can see that the amplitude for all of them is 8. And that b value for all of them is pi divided by 3. The shift, that vertical shift d, we have some that have plus 2, some that have minus 2. Okay. So we can eliminate option b and d because they have the vertical shift being negative, and we found that it's positive too. And then between a and c, we can see that a is gonna be the correct answer because it has that correct c value. Okay. So a is the correct answer here with our function being y is equal to 8 sine of pi divided by 3x minus 5pi divided by 6 plus 2. Thanks everyone for watching. I hope this video helped. See you in the next one.