Tip:
Highlight text to annotate it
X
hello and welcome to Bay colleges intermediate algebra online lectures
in this section uh... we're gonna talk about conic sections initially but then
we're going to concentrate on parabolas we'll see other conic sections
in
the next sections such as nine point three
and nine point four
uh.. what our conic sections well the first thing
that we are going to look at is parabola and we should be relatively
familiar with parabolas
so what i have here is essentially a cone everyone recognizes the shape of a cone
a parabola is a slice of a cone and that's why the sections called conic
sections
if i take a cone
and i cut it
if i can slice it through space
the shape that you would see in there
would be
a parabola
so this is a parabolic shape it's a
conic section a slice of a cone
and it depends on how i slice it if you look at the
if you can see on the camera the angle
uh... at which i cut it creates this
parabola shape
well what if i took the cone and cut it straight across what shape with that
give me
well that would give me a circle
and we already know that a parabola has an equation that describes it in
nine point three
will see that there is actually an equation that describes the shape of a
circle
uh... the next
conic section that we'll look at
is the ellipse and the hyperbola well the ellipse is essentially what if we didn't cut it
straight across what if we cut it
at an angle we cut this cone
like this
what you would get
is an elliptical shape if you can see this red
value here it's kinda like a circle but stretched in one
direction
so we could see that it's not round
it's actually
oval
and that's
called the ellipse and there is an equation
that describes that which will see in section nine point four also in nine
four
we'll see what a hyperbola
a hyperbola you actually have to imagine two cones one cone here and a
mirror image of another cone below it
well if i cut this cone
straight through
i get this
parabola shape
but on the other cone if i cut it all the way down through its mirror image as
well
i would have another parabola
but they would be opening
in opposite directions i'd have one parabola up
and down here i'd have one parabola down that's called a hyperbola
we get this shape here
the other cone would give us that shape over there so these cones are
essentially mirror images of each other
and that's a hyperbola
and there is an equation that describes that
what we'll see in section nine four is the equation of an ellipse and a hyperbola are
very similar
except for one sign change in that sign change just says
just like we saw the previous section sign changes mean reflections well
we're not just looking at a single
piece were actually looking at a piece
and it's reflection cut
if we had a reflecting cone
the next thing we're going to look at
is we are going to concentrate on parabolas
slicing that cone at an angle
so that we just cut through part of it so we get that
parabola shape
now if we recall from chapter seven
a parabola can be written in standard form f of x equals
some coefficient a
times the quality x minus h quantity squared and if we recall from last
section nine one we talked about h and k our horizontal shift and
our vertical shift
all right so this is our standard parabola f_ of x_ a y value
equals
a times the quantity x
minus h and where our y's and x
is for this example is very important
that quantity is squared plus k our vertical shift
if we recall
how we deal with this well our vertex is the value h and k
so i have h comma k
h and k tells me the vertex
a if a is positive it opens up, and if it opens down
a would be negative
our axis of symmetry if you recall from seven point five
is always
x equals
the h value
this is the eqaution
of a vertical line
so let's just say it's our our library function
where h and k are zero
so i have my vertex h and k in this case zero zero
and my axis of symmetry is x equals h well i just said h was zero
so i can see everything's symmetric
from this h value to the right and to the left
and that's why x equals h
and this may be shifted to the left or to the right but x equals h
will always be the axis of symmetry
now what if we have
a cone that we're gonna cut
but it's not
a cone standing
vertically up or down what if it's a cone laying on its side and we cut it
well we can describe that just by changing our axis'
here i have x equals
a times a quantity y minus k and remember k always affects y it's our
vertical shift our y value change
this quantity is squared plus h
so how would i graph this
if we plotted points like we would in any graph well since these are
x's and y's and h and k's we can't really just pick points and plug them in 'cause
there's no math to do here
uh... what we do is we use the same concept well what is the vertex
the vertex is always the point h and k
so we have to apply what we
looked at parabolas before
h always
affects x so h is the horizontal shift the x value change
well that goes in here it's still h
but notice the h is actually outside the parentheses so we're treating it as if
it were k
but because of all points are x_y_
the h value still has to go here this is how
x is changing
and then we have the k value our vertical shift but it's in these
parentheses
and if we recall anything in these parentheses
the sign we see here we change it so the k value it's y minus k
the k value still affects y it is a y value so we put it in here our
vertex is h_k_
so let's say
again just like in the last example i said well let's
just imagine for a moment h and k are zero zero
all right what's the axis of symmetry well because this is a parabola
on its side
the axis of symmetry has to do with y
because it's symmetric
because it's laying on a side maybe it opens
to the right a positive a
or to the left
a negative a
so what we have to do here say y equals
the axis of symmetry
y deals with
the k value y equals
k is our axis of symmetry
so essentially we have to realize because
the x's and y's have changed positions
um... that's essentially what we had covered in chapter eight
uh... what what we call the inverse of these are inverses of each other we're
switching our x's and y's
to get the inverse so we switch our
positions of h and k but we never switch position and points they are always
x value y value
we also switch
this so our axis of symmetry instead of being x equals h
it is now y equals k
so be very careful look for where that x is
look for where that y is which of these variables as being squared tells us
what's gonna happen there
and just like when we learned about a in standard form of a parabola if
a is positive it opens up the positive values
if it's negative it opens down the negative values that applies here as
well
if a is positive it opens in the positive direction
or if a is negative it opens in the negative direction
so be very careful when we get to these inverse parabolas because they are laying on their
sides
now notice this is not written in function notation
because this would not pass the vertical line test if I have a parabola
opening to the right
this would not pass the vertical line test this is not a function
but we can still
describe it, we can graph it as a conic section.
Alright let's look at an example of this here
we have x equals negative 2 times of quantity
y plus 3
squared. Now
if I notice this, the first thing I want to do is
determine what is my vertex, what is my h, k?
Well if we do that,
what value is in these parentheses? Well this affects y
so this is a k value
and it's always the opposite of what we see in here before we square it, so the
k
is a negative 3.
Well what's my h value? Well because this is a parabola that's on its side
the h value's out here
well there is no h value. Well that just means
h is 0 so 0 negative 3 is my vertex. Let's go ahead
and graph that x is 0
y is negative 3 so that's my vertex.
0
negative 3
Alright, but this is a parabola
on its side, so does it open to the right or to the left?
Well a my a coefficient here's a negative value
that tells me
that it opens to the negative direction x is negative in this direction
so it's a parabola
that's going to open this way
one other thing we can do check is we can look at that axis of symmetry well
because it's a parabola on it's side the axis of symmetry is a y value
y equals
y equals k in this case we found k to be negative 3
y equals negative 3 if I draw that axis of symmetry
and I'm going to label it y equals
negative 3
a horizontal line
we can see yep it's symmetric above and below that line
so we can graph this just like we did with our normal parabolas the only
difference is they're inverse parabolas they're opening to the left or
to the right
instead of up or down
as we had seen in chapter seven
sometimes our quadratic equations are not in standard form
this is this is our general form
a, y squared plus b, y plus c
and if we recall we could write it into
a form
of h's and k's if we complete the square so let's just refresh
on how to complete the square
The first thing I'm gonna do is I gotta have this value
as a 1 that a coefficient has to be one so i'm gonna divide all the other
terms
uh... by 3 essentially or I'm gonna factor out
let's factor out.
just out of my y terms
so i'm just factoring that little piece out.
Now to complete this square, we take one-half of b
and we square it well half of 2
let's actually show that one-half
of the b value squared
one half of 2 is just 1
1 squared is 1
this is the value I need to add inside these parentheses.
Now if we recall completing the square what we do to one side we have to
do to the other, so I added over there I also have to add a value over here.
This is where we need to be
real careful. It's 3 times the quantity in here well if I added 1 in
here,
I actually change this whole equation by 3 times
what I added in there. So I change it by 3
So I could add 3 to this side of the equal sign
but just to save us some
trouble here what I'm doing is adding 3 to this side.
I'm gonna have to move it back over here eventually so I'm just gonna subtract it
now.
So if I
change this side of the equation by 3, 3 times this value I put in
here,
I can undo that change by subtracting that 3
that way we save a little bit of time of moving things across the equal sign.
Hopefully you've follow that at home.
Now I can do a little bit a simplifying because I did make this a perfect
square, hence the term completing the square.
and i'm just gonna re-write it x equals 3
times this perfect square which is going to be y
and the quantity in here before I square it is what it always
factors to. So that's why I like to show these steps
and then 7 minus 3
is 4
and if we look at this, this is in standard form.
a times the quantity y
minus k squared plus h.
So now I'm ready to
put this all together I can
determine what my h or what my k and my h are
what a is this is going to be a parabola
that's positive
laying on its side because y is the squared value
so it's going to be a parabola that opens to the right
and I know it's vertex is going to be 4
negative 1 and I could graph that
so what if
what is another method that we learned
instead of completing the square while we talk about the vertex formula
if we recall for parabolas it was negative
b
over two a and then the function evaluated
uh... at that value negative b over two a plug it in and see what you get
well because this is an inverse a parabola on its side
inverses we recall switch our x's and y's so my vertex formula for a parabola
on its side opening either left or right
is going to be
the negative b over two a
is going to tell me
what the k value is because we switched them right it's an inverse
and them i'm going to plug this value in
to find h now i can't use function notation because this is not a function
right it wouldn't pass the vertical line test
so i'm just going to plug evaluate the function
for that value
and that'll do uh... give me the vertex
formula if I just
find k and then plug it in
so let's just do that lets do negative b over two a of this original function
here
negative b would be negative six over two times a
so
uh... six over two times three is
uh... six so negative six over six is negative one i've just found the k
value to be negative one
if we look here what was our k_ value it's always the opposite
of what we see in here well that would be negative one so
that's a true statement it worked we found k
because k effects y it is in the y value of our point for our vertex
now h
if i plug into the original equation the k that i found negative one squared
is one times three is three negative one-time six is negative six
plus seven so i'm just evaluating it for that y value of the k we found
three minus six is negative three negative three plus seven is positive
four my h value four negative one and if we look back after completing the square
we got the same value four for our h
negative one for our k so either way you do it you're going to come up with
the same results ok a lot of times the vertex formula
is gonna save you time
but be careful if it's a parabola on its side
you use negative b over two a to find k
plug that into find h
just as we did here
but now that we've done all this work
to find
x equals
three y plus one quantity squared plus four
once it's in this form this is the easiest way to graph it first identify the
vertex so h and k we already determined h is four and k is negative
one so i'm gonna graph this point
i'm gonna go four
to the right and down one
there's my vertex
what is my axis of symmetry well the axis of symmetry for a parabola on its side
is going to be y equals k so my axis of symmetry
is y equals
negative one
and then
i can go ahead and graph this using this axis of symmetry which way does this
parabola open well
my a value is positive
that tells me that opens to the right
so i can now
graph this parabola
to the right and i know it's a little narrower because this coefficient is
greater than one
a lot of the stuff we're working with when it comes to these
deals uh...
with what we covered in seven point five if you struggled in that section or you just
need the review
go back and watch the seven point five video
practice the problems for seven point five and you'll be able to do it
the only difference here is now these parabolas may be laying on their side
it's x equals some y squared value instead of y equalling
some x sqaured value
so this has been conic sections nine point two dealing with parabolas
thank you for watching