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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