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Manipulating quadratic and exponential expressions — Harder example

Video transcript

dinah 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 given that 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 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 are equivalent expressions 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 2 times 50 times W so minus 100 W plus 2500 and then you have plus 2500 out here now this is going to 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 100 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 is going to be well let's look at this first one this first one let's think about how this this expression works if this this X this part right over here W minus 50 squared that's always going to be non-negative because if you take the square 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 want to 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 if the 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 because 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 coefficient the value of the width for which the area is maximum we know that the value of the width for which the area is 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 want to say look this is going the area's 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 is equal to 0 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 and these don't help you so much you can't make that same type of that same type of intuition about where the maximum value is and this form of a quadratic this is called vertex form and makes it very easy to find the vertex which is going to be the minimum or the maximum value of of an expression or of a function I should say now there's other ways to come up with the 50 you could 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 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 could say that that coordinate is going to be halfway in between the W values that make our function equal to zero and so we could 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 is equal to zero so or W is equal to W is equal to hundred if W is equal to hundred 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 halfway 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 and negative B over 2a if you if you were if you had an expression in the form of a w squared plus B W plus C then and then we could say a a is equal to this is a lowercase here then the vertex is going to be the the W coordinate I should say of the vertex is going to be negative B over 2a 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 1/2 times a is negative 2 and negative divided by a negative is a positive hundred divided by 2 is 50 so anyway 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 coefficient well I see a 50 right there it's a little bit of a strange way of asking it but there's a 50 there it's being subtracted but it does show up none of these other ones that you have a 50 anywhere to be seen