Hey guys welcome to another maths lesson
by Magic Monk.  Today we're going to talk
about if you were given a graph of a
quadratic equation how would you find
the equation of the graph so let's draw
a quick quadratic graph
which is in the form of a parabola so
let's say this is the x axis and this is
the y axis and I'm going to give you the
x-intercepts so I'm going to give you x
intercept of 2 and an x-intercept of 5
and I'm going to say that the parabola
passes through these two points and I'm
also going to give you one other point
and that is the y-intercept I'm going to
say it passes through the y axis at 3 so
how would you find the equation of this
graph well if you remember the factored
form of quadratic formula of quadratic
equations where if we have y equals for
example X minus a and X minus B then the
x-intercepts will occur at x equals a or
b right let's try that here so let's try
putting in the graph of y equals x
- 2 and X - 5 into the  and
see what happens let me just clear the
graph so if I put in X minus 2 and X
minus 5 into  graphmatica it gives me
a parabola and the parabola does have
x-intercepts at x equals 2 and x equals
5 so that's great but is that the graph
we want no it's not because we want the
line to cross the y-axis at y equals 3
so we know we're getting close but that
is not the equation so what I'm gonna do
is I'm gonna add a coefficient at the
front of this term and I'm gonna call it
y equals a bracket actually since we
already used a before actually it
doesn't matter
so at y equals a bracket X minus 2 X
minus 5 and we want to find out what a
is by the way this a has nothing to do
with this a it's just another another
variable I've made up so I'm going to
call it another another variable just so
you don't get confused I'm going to call
it C
so what is c once we find out what c
is then hopefully we will end up with an
equation that is that matches the graph
so let's say let's now use the
y-intercept that we provided so this
point here says that when X is equal to
0 Y is equal to 3 that is the
y-intercept of this graph so I'm going
to put that coordinate into the equation
so 3 equals c bracket 0 minus 2 bracket
0 minus 5 so I've substituted this point
into my equation and that way I can find
out what c is so now I'm going to
simplify the equation c times negative 2
times negative 5 so that's 3 equals 10 c
so c equals 3 over 10 so I'm now going
to try typing this equation into graphmatica 
so y equals 3 over 10 X minus 2
X minus 5 and I'm going to see whether
that is going to give me the same graph
as the graph the line that I've drawn
here so y equals let me clear that y
equals 3 over 10 notice I'm putting 3
over 10 in brackets X minus 2 X minus 5
and if you have a look at the parabola
this is exactly the same as what we were
looking for it passes through
the y-axis at 3 and the x-intercept is
that x equals 2 and x equals 5 so I'm
going to give you another one and see if
you can find it so here it is an
equation a graph where the X intercepts
I'm going to say are 4 and negative 2
and it crosses the y axis at negative 1
just to see whether that is possible
maybe it's not possible I'm not sure so
have a go see you can find it pause the
video come back once you've had a go
okay so the equation is y equals C
bracket X minus a bracket X minus B so
we know that a and b are negative 2 and
4 so c bracket X minus negative 2 and X
minus 4 and here obviously we can rub it
out and make it into a plus here because
minus a negative becomes a plus okay so
what we want to do now is find C so when
x equals 0 y equals negative 1 so
negative 1 equals C bracket 0 plus 2
bracket 0 minus 4 negative 1 equals C
times 2
times negative 4 c equals sorry  2 times 4 is
negative 8 negative 1 equals C times
negative 8 so C equals negative 1
divided by negative 8 which is just 1
over 8 so C is equal to 1 over 8 so
putting that into our final equation we
should have this line as y equals 1 over
8 bracket X plus 2 bracket X minus 4 so
let's try graphing this in graphmatica
so I'm gonna get rid of my other
equation so I'm going to type in y
equals 1 over 8 X plus 2 X minus 4 and
it has indeed given me a parabola where
it touches the x axis at negative 2 & 4
and touches the y axis at negative 1 so
that's awesome so hopefully now you guys
can find the equation of a parabola
given the x-intercepts and one other
point in this case the y intercept see
you next time.
