Vertex & axis of symmetry of a parabola

we defined the vertex and the axis of symmetry of this graph and we're going to I mean the whole point of doing this problem is so that you understand what the vertex and axis of symmetry is and just as a bit of a refresher if a parabola looks like this the vertex is the lowest point here it's this minimum point here for an upward-opening parabola if the parabola opens downward like this the vertex is the topmost point right like that it's the maximum point and the axis of symmetry is the line that you could reflect the parabola around and it's symmetric so that's the axis of symmetry that is a reflection of the left-hand side along that axis of symmetry same thing if it's a downward-opening parabola and the general way of telling the distri in an upward opening and a downward opening parabola is that this will have a positive coefficient on the x squared term and this will have a negative coefficient and we'll see that in a little bit more detail so let's just work on this now in order to figure out the vertex there's kind of a quick and dirty formula but I'm not going to do the formula here because the formula really tells you nothing about how you got it but I'll show you how to apply the formula at the end of this video if you G from you know if you see this on a math test and just want to do it really quickly but we're going to do it the slow intuitive way first so to find let's think about how we can find either the maximum or the minimum point of this parabola so the best way I can think of doing it is to complete the square and it might seem like a very foreign concept right now but let's just work do it one step at a time so I can rewrite this as y is equal to well I could factor out a negative two it's equal to negative two times x squared minus four X minus four and I'm going to put the minus four out here and this is where I'm going to complete the square now what I want to do is Express the stuff in the parenthesis as a sum of a perfect square and then some number over here and I have x squared minus 4x if I wanted this to be a perfect square if I wanted this to be a perfect square be a perfect square if I had a positive four over here if I had a positive four over there then this would be a perfect square it would be X minus two squared and I got the four because I said well I I want whatever half of this number is so half of negative four is negative two let me square it that will give me a positive for right there but I can't just add a four really nearly to one side of an equation I either have to add it to the other side or I would have to then just subtract it so here I haven't changed the equation I added four and then I subtracted four I just added zero to the two this little expression here so it didn't change it but what it does allow me to do is express this part right here this part right here as a perfect square x squared minus four X plus four is X minus two squared it is X minus two squared and then you have this negative two out front and then you have this negative two out front multiplying everything and then you have a negative four minus negative four minus eight just like that so you have y is equal to negative two times this entire thing and now we can multiply out the negative two again so we say we could distribute it Y is equal to negative two times X minus two squared and the negative two times negative eight is plus plus sixteen now all I did is algebraically rearrange this equation but what this allows us to do is think about what the maximum or minimum point of this equation is so let's just look at let's explore this a little bit this quantity right here X minus two squared if you're squaring anything this is always going to be a positive quantity that right there is always always positive but it's being multiplied by a negative number so if you look at the larger context if you look at the always positive multiplied by the negative two that's going to be always that's going to be always negative and the more positive that this number becomes the more positive this number becomes when you multiply it by negative the more negative this entire expression becomes the more negative this entire expression becomes so if you think about it this is going to be a downward-opening parabola we have a negative coefficient out here and the maximum point on this downward-opening parabola is when is when this expression right here is as small as possible if this gets any larger Skitz multiplied by a negative number and then you subtract it from 16 so this x pression right here is zero then we have our maximum y-value which is 16 so how do we get X is equal to 0 here well the way to get X minus 2 equal to 0 so let's just do it X minus 2 is equal to 0 so that happens when X is equal to 2 so when X is equal to 2 this expression is 0 0 times a negative number it's all 0 and then Y is equal to 16 y is equal to 16 this is our vertex this is our maximum point we just reasoned through it just looking at the algebra that the highest value this can take on is 16 as X moves away from 2 in the positive or negative direction this quantity right here it might be negative or positive but when you square it it's going to be positive when you multiply it by a negative 2 is going to become negative and it's going to subtract from 16 so our vertex right here is X is equal to 2 X is equal to 2 actually let me let's say each of these units are 2 so this is 2 this is 2 4 6 8 10 12 14 16 so my vertex is here that is the absolute maximum point for this parabola and it's axis of symmetry is going to be along the line X is equal to 2 along the vertical line X is equal to 2 that is going to be its axis its axis of symmetry and now if we're just curious for a couple of other points just because we want to plot this thing we could say well what happens when I don't know what happens when X is equal to 0 that's an easy one when X is equal to 0 Y is equal to 8 so when X is equal to 0 we have 1 2 3 4 these are 2 2 4 6 8 it's right there this is an axis of symmetry so when X is equal to 3 y is also going to be equal to 8 so this parabola is going to be it's a really steep and narrow one that looks something like this where this right here is the maximum point now I told you this is kind of the slow and intuitive way to do the problem if you wanted a quick and dirty way to figure out a vertex there is a formula you can derive it actually doing this exact same process we just did but the formula for the vertex is x is or the x-value of the vertex or the axis of symmetry is X is equal to negative B over 2a so if we just apply this but you know this is just kind of mindless application of a formula I wanted to see show you the intuition why this formula even exists but if you just mindlessly apply this you'll get what's B here so X is equal to negative B here is 8 8 over 2 times a a right here is a negative 2 2 times negative 2 so what is that going to be equal to it is negative 8 over negative 4 which is equal to 2 which is the exact same thing we got by reasoning it out and when X is equal to 2 y is equal to 16 same exact result there that's the point 2 comma 16