Understanding equivalent forms of rational functions to answer questions in context

0:00- [Instructor] "The company uses a quadratic function to model its profit in dollars. This function represents the profit from selling q items and can be written in three equivalent forms." All right, they just algebraically manipulate this. This is standard form.

0:13If you factor it, you get factored form. And then you could do different types of manipulation to do it in vertex form, essentially some variation. You could think of kind of like completing the square. "Which is the most convenient for determining break-even quantities?"

0:27So pause this video and have a go at it before we do it together. So if we're thinking about break even, break even, it means when your profit has... You're not losing money anymore. You are at break even. So another way to think about it, break even is when P of q is equal to zero.

0:45So you're trying to find the qs that make your function equal to zero, or in other words, you're trying to find the zeros. And if I'm trying to find the zeros, they really jump out in factored form. I can just look at it here and say, okay, anything that makes this expression zero or that expression zero is a zero.

1:02So q is equal to two, or q is equal to positive 10. So factored form is definitely useful for that one. Any of these others, I would actually just try to get it in... Well, the first one I would try to get it into factored form. So that's work. And in the vertex form, I would subtract 16 from both sides, and then I would try to take the positive and negative square roots.

1:27So there's more work involved. So, definitely factored form. Let's do another example here. So let me close these and bring my other example into it. "Consider the graph of the function y is equal to," and we have all of this business in the numerator. Let me scroll down a little bit so you can see this a little bit better.

1:48"Does this form provide the fewest algebraic steps to determine the graph's horizontal asymptote?" So if I wanna look at horizontal asymptotes, I would rather the numerator and the denominator each be in standard form because what I'm going to do is think about, okay, when x gets very, very large, very positive or very negative, which terms dominate?

2:11And here it's hard to tell that. I want it in the form of, in the numerator here, it's going to be x squared plus something times x plus something else, all of that over... Well, it looks like over here, we're gonna have x to the third plus something else times x squared plus something.

2:31I won't use e. Plus, I don't know, I'll call it... I'll use f, although f isn't necessarily the best letter, but you get the picture, f of... Well, that's why I shouldn't use f, d maybe. Oh, I could use d, d times x, and then I could use... My dog just walked in, and the piano is outside, so hopefully y'all aren't getting too distracted 'cause I am.

2:52And I could keep going, and then I would see what happens when we get to very, very large xs. Now when I started to expand it out here, I can see that the denominator is going to grow much, much faster, and so this is going to approach zero. But I had to do all of that work.

3:07If ideally it was already in this form, I could have just done it very quickly. So this is not the fewest algebraic steps, and let's see. "No, to find the horizontal asymptote, we need to continue to factor the denominator." That's not right. The denominator is already...

3:21Well, it's pretty factored, so we would not rule that one out. "No, to find the horizontal asymptote, we need to multiply the factors in the numerator and the denominator in order to compare the degrees." Yep, that's kind of what I tried to do. And we are done.