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.