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# Parabolas intro

Graphs of quadratic functions all have the same shape which we call "parabola." All parabolas have shared characteristics. For example, they are all symmetric about a line that passes through their vertex. This video covers this and other basic facts about parabolas.

## Want to join the conversation?

- does the parabolas ever "touch" and turn into an ellipse?(10 votes)
- I assume that you are talking about the lines of a parabola. If so, no. because then what you are looking at is not a parabola. It would be, as you said, an ellipse or circle.(24 votes)

- Does the opening upwards or opening downwards of a parabola depend on anything?(16 votes)
- Good question. If you are using an equation for a parabola in the form of y=ax^2+bx+c then the sign of a ( the coefficient of the squared term ) will determine if it opens up or down.

Sal has a video 'Introduction to parabola transformations' https://www.khanacademy.org/math/algebra/quadratics/transforming-quadratic-functions/v/shifting-and-scaling-parabolas and addresses this at3:50

There are some diagrams here : http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php(31 votes)

- After a month, finally got to unit 14 (26 votes)
- I dunno, parabolas seem to be a bit more curved than very straight forward...(21 votes)
- What would the equation be for a Parabola that is sideways?(12 votes)
- x=y^2 would create a parabola that goes sideways. Note: it would not be a function.(11 votes)

- I'm here fixing what I missed in school but now I got to something we didn't even study yet I can only feel proud(14 votes)
- Why are they called "quadratics"?(3 votes)
- The Latin word for square is
*quadratum*. Since the area of a square is the square of its side length, words similar to this are sometimes used to mean square, the operation. Quadratics are named thus because the function involves squaring the input.(9 votes)

- Is it possible to determine the magnitude of the curve of a parabola ?

Like line graphs have slopes which shows the magnitude of their slope-iness (i can't think of a better word :P ). So is there anything like that for parabolas ? And how do you find it ?(3 votes)- Yes, by comparing it to the parent function, y = x^2. On a graph, the parent function has the vertex at the origin (0,0) and additional reflexive points (1,1) and (-1,1) because both (1)^2 and (-1)^2 equal 1, then (2,4) and (-2,4), (3,9) and (-3,9). So if we go over 1, we can see how much we go up to see the magnitude. If we start at the vertex (it does not matter where it is on the graph), go over 1 and count how much you go up or down to determine the magnitude. Several examples and for simplicity's sake, keep the vertex at the origin. If I go over one up two, then the equation is y = 2x^2. over 1 up 3 it is y = 3x^2, over 1 down 1, then y = - x^2, over 1 down 2, then y = -2x^2. It gets a little harder with fractions, but if I go over 2 up 2, then it is y = 1/2x^2 (compared to (2,4)) or over 2 up 1, then it is y = 1/4 x^2, over 2 down 3, then it is y = - 3/4 x^2, over 3 up 3, then it is y = 1/3 x^2.(9 votes)

- Why exactly does the second degree polynomial have a parabola for it's graph? The only thing I could think of is example, f(x)=x^2 turns out to be a curve because x can have both n and -n for it's value as it's a square (eg. 2 and -2) so the line goes to both sides of the graph and because the rate of change is exponential and not linear(4 votes)
- the rate of change isn't exponential. since it's 2nd degree, the rate of change of the rate of change is constant. if it was cubic, then the rate of change of the rate of change of the rate of change is constant. also bc of 2nd degree, the graph changes direction once, so possibly 2 values of x can give the same value for f(x). if it were cubic, the graph would change direction a maximum of 2 times, and possibly 3 values of x can give the same f(x). hope this makes sense(4 votes)

- Can a parabola be sideways?(1 vote)
- Yes. If a parabola is sideways, x is equal to y^2, instead of the other way around. Note that a sideways parabola isn't a function, though.(9 votes)

## Video transcript

- [Instructor] In this video,
we are going to talk about one of the most common types
of curves you will see in mathematics, and that is the parabola. 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 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 gonna describe it a little bit more algebraically. In this one, we just wanna get a sense for what parabolas look
like and introduce ourselves to some terminology around a parabola. 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 opened
upwards like this yellow one and this pink one, and some
of them are open downwards. You will hear people say
things like open, opened down, open downwards or open
down or open upwards, so it's good to know what
they are talking about, and it's, hopefully,
fairly self-explanatory. Open upwards, the parabola is open towards the top of our graph paper. Here it's open towards the
bottom of our graph paper. This looks like a right-side up U. This looks like an upside
down U right over here. This pink one would be open upwards. Now another term that
you'll see associated with the parabola, and
once again, in the future, we'll learn how to calculate these things and find them precisely, is the vertex. 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 if 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. It looks like the y
coordinate, it looks like it is about negative three and a half. Once again, once we start
representing these things with 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. In general when we're talking
about, well not just three, two dimensions but even three dimensions, but especially in two dimensions,
you can imagine a line over which you can flip the graph, and so it meets, it folds onto itself. The axis of symmetry for this
yellow graph right over here, for this yellow parabola,
it would be this line. I'm gonna have to draw
it a little bit better. 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'd say that line is x is equal to 3.5. Similarly the axis of symmetry
for this pink parabola, it should go through the
line x equals 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, 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 graphed lines, they're saying where is 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 zero comma three, zero comma three. The y-intercept for the pink
one is right over there. At least on this graph paper,
we don't see the y-intercept, but it eventually will
intersect the y-axis. It just will be way off of this screen. 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 do you intercept or intersect the x-axis? 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 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 one comma zero and six comma zero. 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 are both exactly two and a half 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. 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. 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. 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 two x squared minus five x plus seven. These types that we'll talk
about in more general terms, these types of equations
sometimes called quadratics, they are represented,
generally, by parabolas.