Need up to 30 seconds to load.
in this video we're gonna focus on
finding the vertex of a parabola of a
quadratic equation given in standard
form vertex form and factored form so
what is standard form it's this form Y
is equal to ax squared plus BX plus C so
to find the vertex you need to use this
equation which will give you the x
coordinate of the vertex negative B over
2a
now let's eat your x value you can plug
this back in to the original equation to
get the Y value so that's what we're
gonna do right now so let's say if we
have this quadratic equation y is equal
to x squared minus 4x plus 3 now because
this is positive x squared we know the
parabola is going to open upward and the
vertex is basically this point right
there so let's find the x-coordinate of
the vertex so what's a and what's B a is
the number in front of x squared if you
don't see a number it's a 1 B is the
number in front of X which is negative 4
and C is 3 but we don't need that in
this example so we know that X is
negative B over 2a so that's a negative
and then B is negative 4 divided by 2
times a which is 1 negative times
negative 4 is 4 and 4 divided by 2 is 2
so now we have the x coordinate of the
vertex now we need to find a
y-coordinate so all we're gonna do is
we're gonna replace X with y in that
equation so Y is equal to 2 squared
minus 4 times 2 plus 3 2 squared is 4 4
times 2 is 8 now 4 and minus 8 is
negative 4 and negative 4 plus 3 is
negative 1 so the vertex is 2 comma
negative 1 so that's how you could find
it given an equation in standard form
now let's try another example so let's
say if we have the equation y is equal
to negative 2x squared plus 8x minus 5
so feel free to pause the video and work
on this problem so a is negative 2 and B
is equal to 8 so let's use the same
formula X is equal to negative B divided
by 2a so that's going to be negative
times positive 8 over 2 times a where a
is negative 2 so we have negative 8
divided by 2 times negative 2 which is
negative 4 and negative 8 divided by
negative 4 is positive 2 so we have the
x-coordinate vertex now let's find the
y-coordinate by plugging in back into
the original equation so y is equal to
negative 2 times 2 squared plus 8 times
2 minus 5 now 2 squared is 4 8 times 2
is 16 negative 2 times 4 is negative 8
and 16 minus 5 is 11 negative 8 plus 11
or 11 minus 8 that's positive 3 so the
vertex is 2 comma 3
now try these two y is equal to 4x
squared minus 12x and also y is equal to
2x squared minus 9 so for the example on
the left a is 4 and B is negative 12 so
using the formula X is equal to negative
B divided by 2a that's gonna be negative
times negative 12 divided by 2 times 4
so that's 12 divided by 8 which we can
reduce in so 12 is basically 4 times 3
and 8 is 4 times 2 so you can cancel a 4
so the x-coordinate the vertex is 3 over
2 it's a fraction so now let's find a
y-coordinate so it's gonna be 4 times 3
over 2 squared minus 12 times 3 over 2
now 3 over 2 squared if you square 3 3
squared is 9 2 squared is 4 now 12 times
3 is 36 and 36 divided by 2 is 18 and 4
times 9 over 4 the 4s will cancel so you
can get 9 so this is gonna be 9 minus 18
which is equal to negative 9 so that's
the vertex for the first example on the
right
now let's focus on this example so
what's a what's B and what's C a is 2 B
is the number that I mean C's the number
that doesn't have an X variable next to
it so C is negative 9
but what's B in this problem if you
don't see an X variable then there's
nothing be a 0 so in this example it's
gonna be negative 0 over 2 times 2 0
divided by anything except 0 is going to
be 0 so this negative 0 divided by 4 is
just 0 so now that we have the x
coordinate the vertex we could find the
y coordinate so it's gonna be 2 times 0
squared minus 9 which is 0 minus 9 and
that's negative 9 so for this example
the vertex is 0 negative 9 so that's
what you can do if you see a quadratic
equation in that form
now the next form we need to talk about
is the vertex form which is y is equal
to a times X minus H squared plus K so
for this one you really don't have to do
any calculation you could just look at
it and see that the vertex is gonna be
the vertex is H K so here you see
negative H and here you see positive H
so whatever number you see here you're
going to have to switch it here is
positive K and this is positive K so you
don't have to switch that number so I'm
gonna give you a lot of examples so you
can understand what to do
let's say if Y is equal to 3 times X
minus 2 squared plus 4 what's the vertex
so whatever number you see here we see
negative 2 just reverse it so this is
gonna be positive 2 and don't reverse
the number that you see here so it's
gonna be 2 comma 4 and that's it that's
how you can find a vertex if you're
given a quadratic equation in vertex
form so I'm gonna write out a few
examples and in the meantime go ahead
and write the coordinates of the vertex
so the next one we're going to switch
the four to negative four and this is
going to stay the same as negative five
so it's negative four and negative five
for the third one we're going to reverse
negative 3 to positive three and the
negative six to K value and we're not
going to change it and here H is
negative three and K is positive 7 so
now you know how to find the vertex of a
quadratic equation in vertex form now
what do you do if you're given a
quadratic equation in factored form how
do you find the coordinates of the
vertex I'm gonna show you two ways the
first way you probably haven't seen
before but maybe you have finding
x-intercepts if you see an X minus 2
factor the x intercept is going to be
positive 2 and for this factor X minus 4
the x intercept is 4 now if you were to
graph it we would have a point here and
another one here and because this is
positive x squared if you were to foil
it the graph has to look something like
this
it turns out that the vertex the x
corner of the vertex is basically the
midpoint of the x-intercepts so if the
x-intercepts are two and four the
x-coordinate has to be the average of
two and four so it's going to be three
now to tively you can see it but let's
say if you get two numbers and you're
not sure what the midpoint is just add
up the two numbers and divide by 2 so 6
divided by 2 is 3 so we have the
x-coordinate of the vertex
now we need to find the y-coordinate so
we just got to plug it into this
equation this is gonna be 3 minus 2
times 3 minus 4 3 minus 2 is 1 3 minus 4
it's
negative one so the vertex is 3 negative
1 now let's get the same answer using
another method it might be longer but at
least we can confirm this answer so I'm
just going to be write the answer that
we have the vertex is 3 1 or 3 negative
1 action now what we need to do is meet
a foil x times X is x squared and x
times negative 4 that's a negative 4x
and then we have negative 2x a negative
2 times negative 4 that's positive 8
negative 4x minus 2x is negative 6x so
now we have the equation and standard
form so we can see that a is 1 because
there's no number in front of x squared
and B is the number in front of X which
is negative 6 so using the formula
negative B divided by 2a
that's going to be negative times a
negative 6 over 2 times 1 so that's
equal to positive 6 divided by 2 which
is 3 and we can see we have that answer
here so now let's plug it in to the
equation so it's gonna be 3 squared
minus 6 times 3 plus 8 3 squared which
is 3 times 3 that's 9 6 times 3 is 18
and 9 minus 18 is negative 9 negative 9
plus 8 is negative 1 and so that gives
us that point so both methods work you
just have to pick and choose which
method you prefer but let's work on
another example so let's say if we have
the formula y is equal to negative 2
times X minus 2 times X minus 5 go ahead
and try this one so using the first
method we have the X intercepts as 2 & 5
so what's the midpoint of 2 & 5 it's
gonna be a decimal 3 point 5 now what
you could do is you can add the two
numbers and divide by 2 so the midpoint
as
fraction is seven over two or three
point five so that's the x-coordinate of
the vertex
so now we got to find a y-coordinate of
the vertex
so let's use the original equation it's
going to be negative two and then times
7 over 2 minus 2 and then times 7 over 2
minus 5 now let's get common
denominators this is 2 over 1 if I
multiply this by 2 over 2 that's
equivalent to 4 divided by 2 and if I
multiply 5 over 1 by 2 over 2 its
negative 10 over 2 7 minus 4 is 3 7
minus 10 is negative 3 so now I can
cancel a 2 so I have negative times 3
times the negative 3 so that's going to
be positive 9 divided by this 2 so the
vertex is 7 over 2 comma 9 over 2 if you
want decimal values that's three point 5
comma 4 point 5 now let's get that
answer using the other technique
so let's begin by foiling X minus 2 and
X minus 5 so we have x times X which is
x squared and then x times negative 5 so
that's negative 5 X and this is negative
2 X and finally negative 2 times
negative 5 which is positive 2 so next
let's combine like terms negative 5 X
minus 2 X is negative 7 X now the last
thing we need to do is distribute the
negative 2 so first we're gonna have
negative 2 x squared and then it's
negative 2 times negative 7 X which is
positive 14 X and then negative 2 times
10 which is negative 20 so now we have
the equation in standard form so let's
find the vertex negative B over 2a so we
can see that B is positive 14 and a is
negative 2
now 14 we can break that into seven
times two so notice that we can cancel a
two negative seven divided by a negative
two is the same as positive seven over
two so as you can see we have the
x-coordinate of the vertex
seven over two is equal to three point
five so let me just make some space here
and I'm just gonna rewrite this formula
on top
so now let's plug in this number into
this equation so it's gonna be negative
2 times 7 over 2 squared plus 14 times 7
over 2 minus 27 squared is 49 and 2
squared is 4 but I'm gonna leave it as 2
times 2
now instead of multiplying 14 by 7 it's
easier if you divide first 14 divided by
2 is 7 and then take that 7 multiplied
by this 7 that's gonna give you 49 so 14
times 7 over 2 is 49 next I'm going to
cancel it to some left over with
negative 49 divided by 2 and 49 minus 20
is positive 29 which I'm gonna write as
29 over 1 now we need to get common
denominator so I'm gonna multiply this
fraction by 2 over 2 so it's negative 49
over 2 plus 20 9 times 2 20 times 2 is
40 9 times 2 is 18 when you add them
that's 58 so negative 49 plus 58 or 58 -
49 that's 9 so this is not over 2 which
we do have that here 9 over 2 is the
same as 4.5 so both ways work you just
have to pick which way is easier
personally I think the first method is a
lot easier than a second but you can
choose so now you know how to find the
vertex of a parabola given a quadratic
equation in standard form the vertex
form and factored form thanks for
watching
