- [Phil] In this video,
we're going to take a quadratic
and find its characteristics
algebraically.
So, here's our quadratic,
minus four X squared
plus 10 X plus nine.
And we're gonna find the following.
We're gonna find the vertex,
the axis of symmetry,
the domain and range,
the X intercepts and the Y intercepts.
I'll have a second video
that will do them all
with a graphing calculator
so that way we can do them both ways
but I'll do them in separate videos.
So, first, let's find the vertex.
Now, remember what the vertex is.
It's the turning point of the parabola.
If it opens upwards,
it's at the bottom,
if it opens downwards, it's at the top.
And we have what I like
to call the vertex formula
which is that X equals minus B over two A.
If you wanna see where this comes from,
look at the video on deriving
the quadratic formula.
So, negative B over two A
is gonna give us the X
coordinate of the vertex.
So, we have to identify
B and A in our quadratic.
Well, B is 10,
A is negative four,
so to find the vertex,
what we're gonna do is apply this,
so negative B over two A,
so negative 10 over
two times negative four
equals negative 10 over
negative eight or 5/4.
Now, a vertex has two coordinates.
It's a point.
The X coordinate is 5/4.
The Y coordinate we can find
by plugging in our X coordinate
to see what we get.
Let's go ahead and use
the calculator for this
so we can see how it would be put in.
So, negative four
times 5/4 squared
plus 10 times 5/4 plus nine.
And I put something in wrong.
I had one extra set of parentheses there.
15.25 or to put it in a fraction
you can use math fraction,
61/4 or 15 1/4.
So, we put in that fourths,
we'll put this one in fourths, 61/4.
So, this is our vertex.
5/4, 64/4.
Next we have what's called
the axis of symmetry.
Now, the axis of symmetry
is the vertical line
that makes our quadratic symmetric.
In other words, if we were to flip it over
that vertical line it would
look exactly the same.
Notice that it goes
directly through the vertex,
so the axis of symmetry is a vertical line
and its equation is just
equal to the X coordinate
of our vertex,
so it's the equation X equals 5/4.
So, once we found the vertex,
we've already got the axis of symmetry,
so not only does this
help us find the vertex,
it finds the axis of symmetry.
Now we are asked to find
the domain and range.
Unless we're in some context,
the domain of all quadratic functions
is negative infinity,
to infinity.
All real numbers will
work for a quadratic.
Now for the range.
Notice that they turn around.
So, let's think about what our quadratic
is gonna look like.
We know that we have a vertex
at one 1/4 and 15 1/4,
so somewhere up here.
So, the next question we ask ourselves
is does our quadratic open up or down?
Remember, we look at
the A to determine that.
The coefficient of X squared.
It's negative so it'll open downwards.
So, our range is gonna be
from this Y value down.
And we found that Y value
as part of our vertex.
So, the range is gonna
be from negative infinity
up to the Y value of our vertex
which is 61/4
and we are gonna want to include that.
So, we can always find our range
from the Y coordinate of our vertex.
It's either gonna be
from the Y coordinate up
or from negative infinity
up to our Y coordinate
depending on which way
our parabola's opening.
All you gotta do again
is look at the A value.
Next thing is our X intercepts.
Because our vertex was above the X axis
and it opened downwards,
we see that we have two of them.
So, let's see if we can
do this algebraically.
We wanna find when negative four X squared
plus 10 X plus nine equals zero.
Now, always try factoring first
but a fall back is always
the quadratic formula.
Let's see if we can get
this thing to factor
into two distinct factors.
So,
we have different ways
we can split this up.
Negative four X can be
split into one and four
or two and two.
Actually let's go ahead and
factor a negative out of that
'cause it'll make it a
little easier to deal with,
so negative 'cause it won't
affect what the zeros are.
Four X squared minus 10 X minus nine
equals zero
and now let's split it
up into two factors.
Again, now four can be
split into two and two
or one and four.
Nine can be split into one and nine
or three and three.
We need the version that's
exactly 10 units apart.
So, in looking at all the
possible combinations,
four times three is 12,
one times three is three,
those are nine apart, that won't work.
Two times three is six,
two times three is six
'cause they have to have opposite signs.
That won't work.
And so, two times nine is
18, two times one is two,
so we're not gonna have a
very easy time factoring this.
So, let's go ahead and
apply the quadratic formula
and see what we get for zeros.
The quadratic formula is as a reminder
minus B plus or minus the square root
of B squared minus four AC
all over two A.
And for our problem,
we can use either version.
Let's go ahead and use our one
that says negative factored out.
So, 10 plus or minus the square root
of negative 10 squared
minus four times four times negative nine
all over two times four.
So, simplifying we have 10
plus or minus the square root.
Here we have 100
plus four times four is 16
times negative four is negative 64,
so plus 64,
so 164
all over eight.
Now, this thing
is not a perfect square
but it does have perfect squares in it.
For instance, it's divisible by four.
If we divide it by four,
we get 41.
Now, in another video I
showed how to simplify this
but we can pull a four out of the radical,
so we get 10 plus or minus two
times the square root of 41
all over eight.
Now we can divide the
top and bottom by a two.
Every term has to get divided by two.
So, that's five plus or
minus square root of 41
all over eight.
And that's a nice simplified answer.
So, those are our two X intercepts,
the plus or minus
accounts for both of them.
Lastly, we're asked for the Y intercepts,
so the X intercepts we'll write as five
plus or minus square root of 41 over four,
let's just say over four, not eight,
comma zero.
The Y intercept is the output
when the input is zero.
This will be pretty easy to calculate
from our original 'cause
if we put in zero for X,
all we get left is the nine
and there's our Y intercept.
So, sorry about factoring.
This is one I thought would factor
but it didn't
but we were able to use
the quadratic formula
to still find our X intercepts.
