Manipulating quadratic and exponential expressions — Harder example

- [Instructor] Dina purchased 200 feet of fencing to make a rectangular play area for her dogs. The possible area A, is given by the equation below where w is the width of the play area. So, they do the work for us, where they tell us what area as a function of w is going to be. Which of the following equivalent expressions displays, as a constant or a coefficient, the value of the width for which the area is a maximum? So that's an interesting way that they phrased it, but they're saying that these are all equivalent expressions to this thing up here. And you could verify that they're equivalent expressions, or we could take, for example, this first one right over here, this would be the same thing as negative, let's see w minus 50 squared is w squared minus two times 50 times w, so minus 100 w plus 2500. And then, you have plus 2500 out here. Now, this is gonna be the same thing as negative w squared plus, I'm just distributing the negative sign, plus 100 w minus 2500 plus 2500, which these two things cancel each other out, and you're left with this expression right over here. Negative w squared, negative w squared plus a hundred w. This would be true for all of these, so these are all just algebraic manipulations of this original thing. Now, what they're asking us, or one way to think about it is, which of these show the maximum value? Which of these forms of the expression make it easy for us to find what the maximum value's going to be? Well, let's look at this first one. This first one, let's think about how this expression works. If this, this part right over here, w minus 50 squared that's always going to be non-negative 'cause if you take the square of something, it's going to be non-negative. But then, if you then take the negative of it, it's going to be non-positive. It's going to be non-positive, so this part over here is going to be less than, or equal to, zero, and then you have plus 2500. So, if you wanted to maximize this expression, which is the same thing as maximizing this expression, because they're equivalent, you want, this thing, you wanna not subtract anything. So, you want this part to be equal to zero. So, how can this part be equal to zero. Well, the thing that you're squaring is equal to zero and, in order for the thing you're squaring to be equal to zero, w minus 50 needs to be equal to zero, or w needs to be equal to 50. So, just looking at this expression, we're able to figure out 'cause all of these expressions are equivalent and they're equivalent to this expression right over here. Just looking at this expression we're able to figure out, look, this expression takes on a maximum value when w is equal to 50. And, to their question which of the following equivalent expression displays, as a constant or a coefficient, the value of the width for which the area is maximum? We know that the value of the width for which the area's a maximum is 50 and this one I guess displays it. I've never seen a question phrased exactly like that. But, it does display the number and this is why this form is so valuable. You wanna say, look, this is going, the area is going to have a maximum value when this part right over here is not taking anything away from the 2500. In fact, 2500 is going to be that maximum area. And, when do we reach it? When this thing is zero, this thing is zero, when w minus 50's equal to zero, or when w is equal to 50. So, I would definitely pick this one here. These other forms of these expressions, and you can algebraically expand these, it'll come out to be the same thing as over here, these don't help you so much. You can't make that same type of intuition about where the maximum value is. And this form of a quadratic, this is called vertex form. It makes it very easy to find the vertex, which is going to be the minimum or the maximum value of an expression, or of a function I should say. Now, there's other ways to come up with the 50. You can say, look, this thing right over here, you have a, it's a quadratic, it's got a negative coefficient on the quadratic term, or on the second degree term, so it's going to be a downward opening parabola. So, it might look something like this where if this is the a-axis, the area axis, and this right over here is the width axis. That's the width axis. You could, you would know that this is gonna be downward opening because it has this negative right over here. It's gonna be a downward opening parabola. So, it's gonna look like this. And so, we just need to find the vertex, or we need to find the w coordinate of the vertex there, and there's a couple of ways that you can find the w coordinate of the vertex. One, you can say that that coordinate is going to be half way in between the w values that make our function equal to zero. And so, we can say, well, what makes this thing equal to zero. You could say 100 w minus w squared is equal to zero. You could factor out a w, so w times 100 minus w is equal to zero. Well, this is going to be equal to zero when either w is equal to zero or 100 minus w's equal to zero. So, or w is equal to, w's equal to 100. If w's equal to 100, this part over here is going to be equal to zero. The whole area is going to be equal to zero, so this right over here is 100. This one over here is zero. The maximum point then, the vertex, is going to be the w that's half way in between these two, which is going to be 50. So, that's another way to think about it. You could also use the formula for the x-coordinate of a vertex, negative b over 2 a. If you had an expression in the form of a w squared plus b w plus c. Then, and then we could say a is equal to, this is a lowercase here, then the vertex is going to be, the w-coordinate I should say of the vertex, is gonna be negative b over two a. And, in this case, in this case, b is 100, it's the coefficient on the first degree term. So, it's going to, if b is 100, negative b is negative 100. And, a is this negative one that's in front of the w squared. It's the coefficient on the second degree term. This, they wrote it the other way around where they have the first degree term and then the second degree. Here, you have the second degree, then the first degree. But, a is negative one. Two times a is negative two. Then, negative divided by a negative is a positive. 100 divided by two is 50, so any way you look at it, w equals 50 is when it hits the maximum value. And, if you looked at these expressions, which of these involve 50 as a constant or a coefficient? Well, I see a 50 right there. It's a little bit of a strange way of asking it, but there is a 50 there. It's being subtracted, but it does show up. None of these other ones have a 50 anywhere to be seen.