Hey guys in this tutorial we're going to
talk about in a parabola if I give you
the Y coordinates can you give me the x
coordinate of the graph so let's have a
look at this graph which we draw
previously in graphmatica and the
equation was y equals x squared plus 5x
plus 6 and let's say I wanted to know
that when y was equal to 2 what's X now
let's have a look at this graph and
let's have a look at where y equals 2 is
so this is the y-axis and y is equal to
2 will be about here so when y is equal
to 2 X is equal to either negative 4 or
negative 1 so there are two answers so X
is equal to negative 4 or negative 1 now
let's try another number when y is equal
to 4 what does X equal 2 so now let's go
to where Y is 4 on the graph when y is
equal to 4 again there are two answers
for when y is 4 x is equal to about
negative 0.5 according to our graph and
there's another answer about negative 4
point 5 so you can see that when I give
you a y-coordinate you can actually give
me two answers for the x-coordinate
because it's a parabola
the only difference would be the turning
points where there is only one
x-coordinate for that Y value for that
y-coordinate
okay so as you can see it's quite hard
to use a graph to give you an exact
accurate number right negative 0.5 I
pretty much estimated that from just
looking at the graph I didn't I don't
have any solid proof other than my eyes
to say that that is the accurate value
so we need to know how to do this
algebraically well let's try the easiest
one
let's make y equals to 0 so when y
equals 0 what's X and according to our
graph when y is equal to 0 which would
be here vertically when y is equal to 0 we
have X is negative 2 and negative 3 so
how we do how we solve that
algebraically is if we get the equation
of the graph y equals x squared plus 5x
plus 6 and we make y equal to 0 we
substitute 0 into the equation for y so
0 equals x squared plus 5x plus 6 the
next thing you need to do is factorize
this equation and if you're not sure how
to do that there's another video in
the year eight to ten section that
teaches you how to factorize a quadratic
equation so basically by factorizing we
have this equals x plus two in brackets
and X plus three in brackets once we
have factorized it then using the null
factor theorem which says that if a
times B is equal to zero
then either a is equal to zero or B is
equal to zero and this makes sense
because if you think of two numbers that
multiply together to give you zero then
one of them has to be zero so for
example 5 times 0 is 0 or 0 times 10 is
0 and so on so whenever you have two
numbers that multiply to give you 0 one
of them has to be 0 so that means in
this equation here we have the first
term which is x plus 2 multiplied by the
second term x plus 3 so one of them has
to be 0 so either x + 2 is 0 or x + 3 is 0
when X when X plus 2 equals 0 what's X
well we're going to minus 2 on both
sides so minus 2 on both sides the
positive and negative 2 cancel each
other out so we have x equals negative 2
and the same thing happens here we're
going to minus 3 on both sides so x
equals negative 3 so these are your
answers x equals negative 2 and x equals
negative 3 and you can see that it
matches our it matches our X
intercepts x equals negative 2 and
negative 3 on our graph so when y is
equal to 0 x is equal to negative 2 and
negative 3 now what if back into our
graph back to our graph I'm just going
to delete some of this stuff
so back in our graph what if I wanted to
know when y is equal to two what's X now
according to this graph it's about
negative one and negative four okay so
I'm going to write out the equation when
y is equal to two two equals x squared
plus 5x plus 6 right because we've
substituted the number two into our
equation now what you want to do is try
to solve this equation now by making
everything go to the right-hand side so
we have 0 equals x squared plus 5x plus
4 and so what did I do here to make that
happen well basically -2 on both
sides and you can see that 2 minus 2 is 0 6
minus 2 is 4 and then so we have this
equation here and you can factorize it
again so that factorizes into X plus 1 X
plus 4 equals 0 using a null factor
theorem you can see that X plus 1 is
equal to 0 or X plus 4 is equal to 0
solving these two linear equations you
get x equals negative 1 or x equals
negative 4 so that's done so we can see
that this matches our graph again when
when y is equal to 2 X is equal to
negative 1 and negative 4 now this
method only works when the coordinates
are whole numbers so let's try another
example when the coordinates are not
whole numbers so if you have a look here
when y is equal to 3 what's X and when 1
is equal to 3 which is about here X is
not a whole number X is about somewhere
between negative 0.5 and they give one
and here it's somewhere between negative
four and negative 4.5 so we're not sure
what it is but it's definitely not a
whole number for X so what we're going
to do now is again write our equation
out substitute the number 3 into y so 3
equals x squared plus 5x plus 6 and
we're going to rearrange the equations
so that all the terms are on the right
hand side and we have 0 equals x squared
plus 5x plus 3 so I've subtracted 3 on
both sides which made the 3 go under
which may turn into 3 and 0 okay now we
want to solve this equation using the
quadratic formula which says that the
quadratic formula says that when x sorry
when ax squared plus BX plus C is equal
to 0 then X is equal to negative B plus
or minus b squared minus 4ac under a
square roots over 2a now this formula is
actually quite easy to memorize after
you've used it a few times so you just
have to memorize that one unfortunately
ok so we have in our equation a is equal
to 1 because a is in front of the x
squared b is equal to 5 because b is in
front of X and C is equal to 3 so
solving that equation we now need to put
A B and C
into our quadratic formula so B is a 5
plus -5 squared minus 4 times 1 times 3
because a is 1 and C is 3 over 2 times 1
so putting all that into the quadratic
formula and let's type so let's get the
numbers and under the square roots are
calculated so 5 squared is 25 minus 4
times 3 is 12 over 2 so that makes it
negative 5 plus or minus square root of
13 over 2 so putting that in the
calculator we have negative 5 plus
square root of 13 over 2 or negative 5
minus square root of 13 over 2 that's
what the plus and minus mean because it
means you can either plus or minus so
we're going to do we're going to try
both and see what happens
and so putting that in the calculator
negative 5 so negative 5 plus the square
root of 13 and divided by 2 so that's
negative zero point six nine seven
round it to 3dp or negative five
oops negative five minus the square root
of thirteen divided by two which is
negative four point three well is that
negative four point 302 303 again round
it to three decimal places and then
let's check that against our graph so
when X sorry when y is equal to three X
should be equal to negative zero point
six and negative four point three so
that matches our graph because when X is
equal so when y is equal to three which
is about here X is equal to negative go
down a straight line which is about
there which is about negative six nine
zero point six nine seven and here if
you go down from Y is equal to three X
is about negative 4.303
so it matches again our graph so
basically if you can get y coordinate
there are a couple of things you should
try number one is try and factorizing
now that won't always work when the
answer is not a whole number so if that
doesn't work then you can use the
quadratic formula quadratic formula
okay so thanks for watching the tutorial
on how to find the x-coordinate given
the y-coordinates in a quadratic
equation see you next time.
