Choosing an appropriate domain for a quadratic function

0:00- [Instructor] We're told to describe a reasonable domain for each situation and explain how to modify the graph to show that domain. So we're starting with situation A, a livestream activity. So a school club is hosting a 12-minute livestream announcement. The number of new viewers joining per minute, y, is modeled by, we have this quadratic equation here, where x is the number of minutes after the livestream starts.

0:26So you give me how many minutes after the livestream, I put that for x and I'll tell you how many people are joining per minute. Now, if we just graphed y equals all of this business, we would get this parabola right over here. But when we think about the actual situation, we realize that having negative time does not make sense.

0:48And also that would be, I guess, before the livestream is actually starting and then it's only lasting 12 minutes. So we shouldn't have time that goes beyond the 12 minutes. So if I'm thinking about the domain here, so the domain to me would be that x would be greater than or equal to zero, or zero is less than or equal to x, and which is going to be less than or equal to 12 where x is the time in minutes.

1:17This is the x-axis right over here. Now, what would that graph look like? Well, essentially we would only be showing the part of this graph from here to here. So let me just trace that out. So what I'm doing in blue would now be the graph of the function with this restricted domain.

1:39That makes sense. We would not look at the time beyond 12, so we wouldn't graph anything there, and we wouldn't look at the time below zero. So we wouldn't graph anything there. Let's look at another scenario. So let me scroll. Lemme get rid of this one and scroll down.

2:02So in situation B, theater light setup, we are told a row of stage lights is numbered 1 through 10. The amount of cable, y, in feet needed to connect light x to a box is modeled by. All right, so once again, they have graphed this entire expression, but what would the domain of this actually look like?

2:25And actually lemme do another color here. So let's think about this domain. So domain in this situation, well, the x is the control box number, and those are numbered 1 through 10. I'm assuming it's only the whole numbers 1 through 10 that you can't have box number 1.5 or box number pi.

2:51So I would say x is a whole number, X is whole number. I could say in between 1 and 10 included. Or I could say where one is less than or equal to x is less than or equal to 10. Now, given this restricted domain, what would that look like? Well, I wouldn't have all of these points on this curve.

3:16I would essentially have when x is equal to one, I would have that point, when x is equal to two, that point, x is equal to three, that point. Four, five, six, seven, eight, nine and then 10. So the graph of this function with the restricted domain would only be these little points that I have colored in red.

3:42That would be the ones where you could actually get a input and x value that would give you a y value that makes sense in this situation. So this function should really only be defined for where x is a whole number between 1 and 10, including 1 and 10.