Parabolas intro

in this video we're going to talk about one of the most common types of curves you will see in mathematics and that is the parabola and the word parabola sounds quite fancy but we'll see it's describing something that is fairly straightforward now in terms of why it is called the parabola I've seen multiple explanations for it it comes from Greek para that root word similar to parable you could view of something beside alongside something in parallel bola same root as when we're talking about ballistics throwing something so you could interpret it as beside alongside something that is being thrown now how does that relate to curves like this well my brain immediately imagines well this is the trajectory this is the path that is a pretty good approximation for the path of things that are actually thrown when you study physics you will see the path you'll approximate the paths of objects being thrown as parabolas so maybe that's where it comes from but there are other potential explanations for why it is actually called the parabola it has been lost to history but what exactly is a parabola in future videos we're going to describe it a little bit more algebraically and this one we just want to get a sense for what parabolas look like and introduce ourselves to some terminology terminology around a parabola so these three curves they are all hand-drawn versions of a parabola and so you immediately notice some interesting things about them some of them are open upwards like this yellow one and this pink one and some of them are open downwards and you will hear people say things like open open down open downwards or open down or open upwards so it's good to know what they're talking about and it's hopefully fairly self-explanatory open upwards the the parabola parabola is open towards the top of our graph paper here it's open towards the bottom of our graph paper this is looks like a right-side up view this looks like an upside down right over here this pink one would be open upwards now another term that you'll see associated with a parabola and once again in the future we'll learn how to calculate these things and find them precisely is the vertex and the vertex you should view as the maximum or minimum point on a parabola so if a parabola opens upwards like these two on the right the vertex is the minimum point the vertex is the minimum point right over there and so someone said what is the vertex of this yellow parabola well it looks like the X looks like the x-coordinate is three and a half so it is three and a half and looks like the y coordinate it looks like it is about negative three and a half and once again once we start representing these things with the equations we'll have techniques for calculating them more precisely but the vertex of this other upward-opening parabola it is the minimum point it is the low point there is no maximum point on an upward-opening parabola it just keeps increasing as X gets larger in the positive or the negative direction now if your parabola opens downward then your vertex is going to be your maximum point now related to the idea of a vertex is the idea of an axis of symmetry and in general when we're talking about well not just three them two dimensions but even three dimensions but especially in two dimensions you can imagine a line over which you can flip the you can flip the graph and so it meets it folds on to itself and so the axis of symmetry for this yellow graph right over here for this yellow parabola it would be this line it could draw it a little bit better it would be it would be that line right over there you could fold the parabola over that line and it would meet itself and that line I didn't draw it as neat as I should that should go directly through the vertex so to describe that line you would say that line is X is equal to three point five similarly the axis of symmetry for this pink it should go through the line x equals a negative one so let me do that that's the axis of symmetry it goes through the vertex and if you were to fold the parabola over it would meet itself the axis of symmetry for this green one it should once again go through the vertex it looks like it is X is equal to negative six this is let me write that down that is the axis of symmetry now another concept that isn't unique to parabolas but we'll talk a lot about it in the context of parabolas are intercepts so when people say y-intercept and you saw this when you first graph lines they're saying where does the graph where does the curve intercept or intersect the y-axis so the y-intercept of this yellow line would be right there it looks like it's the point 0 comma 3 0 comma 3 the y-intercept for the pink one is right over there we at least on this graph paper we don't see the y-intercept but eventually will intersect the y-axis it just will be way off of this screen now you might also be familiar with the term x-intercept and that's especially interesting with parabolas as we'll see in the future x-intercept is where you intercept or intersect the x axis and here this yellow one you see it does it two places and this is where it gets interesting lines will only intersect the x axis once but at most but here we see that a parabola can intersect the x axis twice because it curves back around to intersect it again and so for here the x-intercepts are going to be the point 1 comma 0 and 6 comma 0 now you might already notice something interesting the x-intercepts are symmetric around the axis of symmetry so they should be equal distant from that axis of symmetry and you can see they indeed are they're both exactly 2 and 1/2 away from that axis of symmetry and so if you know where the intercepts are you just take you could say the midpoint of the x-coordinates and then you're going to have the axis of symmetry the x-coordinate of the axis of symmetry and the x-coordinate of the actual vertex similarly the x-intercept here looks like it's negative the points are negative seven comma zero and negative five comma zero and the x coordinate of the vertex or the line of symmetry is right in between those two points now it's worth noting not every parabola is going to intersect the x axis notice this pink upward-opening parabola it's low point is above the x axis so it's never going to intersect the actual x axis so this is actually not going to have any x-intercepts so I'll leave you there those are actually the core ideas or the core visual themes around parabolas and we're going to discuss them in a lot more detail when we represent them with equations and as you'll see these equations are going to involve second degree terms so the most simple parabola is going to be y is equal to x squared but then you can complicate it a little bit you could have things like Y is equal to 2x squared minus 5x plus 7 these types and we'll talk about in more general terms these types of equations sometimes called quadratics and they are represented generally by parabolas