In this
video, I'd like to briefly
talk about the quadratic formula
and the vertex formula
as they relate to one another,
or rather as they don't
relate to one another.
Because consistently
on exams and so forth, when I give a
word problem involving
a quadratic, the
most common error
is to use the quadratic formula
when you should be using the vertex
formula, or vice versa,
using the vertex formula when you
ought to be using the quadratic
formula; and I
of course understand the temptation to
just
just dive in to a problem,
I mean especially on a test, you are
being
timed, you don't want to spend a
huge amount of time thinking about
any specific problem;
at the same time this really is an
error that that is caused
by diving into a problem
without giving it any thought,
because the quadratic formula and the
vertex
formula are used for completely
different things.  I think that
if you give it a little thought, you
shouldn't
confuse the two.
So let's just look at a problem,
the kind of problem that could appear
on a homework 
or on a test.
We won't even solve the problem,
let's just give it a little thought.
A ball is thrown upwards from the roof
of
the building; its height
in feet from the ground after
t seconds is given
by this quadratic equation.
When will the ball hit the ground?
And let's just think
about what's being asked here.
Here
is our quadratic,
and we know how to find
two things. We know how to find
the roots of a quadratic
(only one of them makes sense in this
context) and we
want to, sorry, we know
how to find the vertex of
a quadratic;
and we have got
this graph of the time versus the height,
so we can find one of
two things here:
the vertex is where the height
is at its maximum,
we would find this using
the vertex formula,
or this point where the height
is zero,
we would find that using
the quadratic formula.
So what are we being asked? Are we being
asked when the height has reached its
maximum value,
or are we being asked when the height
is zero?
Well, the ball hits the ground
when the height is zero,
so we would ... I'm not going to
go through all the details, because
that's not
really the point of this video,
but we would use
the quadratic formula to find when the
height is zero.
We'd get two solutions and we'd select
the one that makes the most
sense.
Or consider another equation:
the number of breeding pairs of house
sparrows in a city
is quadratically related to the
so-called human
disturbance by this
quadratic equation;
at what level of human disturbance
will there be the most
sparrows? And again, the point of this
video isn't to work through this
problem completely
it's to ask how would we start
on this problem;
so here's what this
graph looks like,
it continues down here
but sparrow population
can't be a negative so
only this part of the
graph makes
sense, and
we know how to find
this,
the vertex,
and we know how to find these
two points, with
the quadratic formula
How would we start this problem?  I mean, we have
two formulas we could use,
which of them are we looking for?
And again, this is just a case of
stopping
and thinking;
the vertex is the maximum
value.  What are we being
asked for?
Well, just for that.
When is the population at its
maximum?
if we use the quadratic formula,
we would find where the population is
zero ,we'd find the levels of
human disturbance that lead to
extinction, that's,
that's practically the opposite
of what we're trying to do!
We're looking for a maximum;
it's given by the
vertex.
So the vertex
formula
it is.
In spite of it being sort of
my job, I don't like to lecture
students, but it always is
very frustrating
to give a problem like this on a test
and then see students use the quadratic
formula, because there's no reason
for that kind of error, that's the kind
of thing that happens
when you start working on a problem
without thinking about it
first.
I hope you'll take this video to heart,
even on a test even when you want to get
through the problems promptly,
I hope you won't just rush in
thoughtlessly but will take
a moment to give the problem
some thought; because if you do,
I think this kind of error will
stop occurring
