- In this video we're
gonna 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.
On the second video that we'll do them
all on the graphing calculator
so that we can do it in 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
on 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 the x equals minus b over two a.
We wanna see where this comes from,
look at the video on driving
the quadratic formula.
So negative b over two a is gonna give us
the x coordinate of the vertex.
So it identify b and a in the quadratic
where b is 10,
a is negative four,
so, to find the vertex
what we're gonna do is apply
this to negative b over two a,
so negative 10
over two times negative four
equals negative 10 over negative eight,
or five fours.
Now, the vertex has two
coordinates, it's a point,
the x coordinate is five fours,
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 where b put in,
so negative four
times five fours
squared plus 10 times
five fours plus nine.
And I put something in wrong,
I wanna an extra set on parenthesis there.
15 point two five, or
to put it in a fraction
you can use math fraction,
61 fours or 15 in a quarter.
So we'll put that in fours,
we'll put this one in fours, 61 fours.
So this is our vertex,
five fours coma 61 fours.
Next we have what's called
the axis of symmetry,
now, the axis of symmetry
is the vertical line
that makes our quadratic symmetric,
now there's a way to flip
it over that vertical line
it looked exactly the
same, notice that it goes
directly through the vertex,
so the axis of symmetry
is a vertical line and
it's equation is just equal
to the x coordinate of our vertex
so it's the equation
x equals five fourths.
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 here we're asked to
find the domain in range,
unless we're in some context,
the domain of all quadratic functions
is negative infinity to infinity.
All real numbers will work for 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 at a quarter and 15 in a quarter,
so it's somewhere up here.
So, next question we ask ourselves is
does the quadratic open up or down?
Remember, we look at the
a to determinate that,
the coordinate of x, the
coefficient of x squared.
Is negative so it will open up 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 fours and we're
gonna want to include that.
So, we're gonna always find our range
from the y coordinate of our vertex,
it's either gonna be
from the y coordinate up,
or for negative infinity
up to a y coordinate
to vent depending on each
way our parabola is opening.
All you gotta do again
is look at the a value.
Next thing is our x interception,
because our vertex was above the x axis
and it open downwards, we
see that we have two of them.
So, let's see if we can
get this algebraically,
we wanna find where negative
four x squared plus 10 x
plus nine equals zero.
Now we're always try factoring first
but a fall back is always
a quadratic formula,
let's see if we can get
this thing to factor
into two distinct factors,
we have different ways
we can split this up,
four has, negative four x can be split
into one in four
or two in two.
Let's go ahead ahead a factor
a negative out of that,
'cause it will make it a
little easier to deal with,
so negative, 'cause it won't
affect were 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 in two or one in four,
nine can be split into one
in nine, or three in three,
we need the version that's
exactly 10 units apart,
so, in looking at the all
the possible combinations,
four times three is 12,
one times three is three,
those are nine apart, that won't work,
three times three is six,
three times three is six,
as they have to have opposite
signs. That won't work.
And so, two times two,
two times nine is 18,
two times 1 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.
Quadratic formula is, as a reminder,
minus b plus or minus
square root of b squared
minus four a c all over two a,
and for our problem we
can use either version,
let's go ahead and use the one
with the negative factor out.
So, 10 plus or minus
square root of negative
10 squared minus
four times four times negative nine
all over two times four.
So as simplifying we have 10
plus or minus square root,
here we have a 100,
plus four times times four,
is 16 times negative fours, 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 is divisible by four,
if we divide it by four we get 41,
now in other videos I
showed how to simply this,
we can pull a four out of the article,
so we get 10 plus or minus
two times square root of 41,
all over eight, now we can
divide the top and bottom by 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 plus or minus
accounts for both of them.
Lastly we're asked for the y intercepts,
so the x intercepts will write as five
plus or minus square root of 41 over four,
let's just say over four
and not eight, coma zero.
The y intercept is the
output when you input a zero,
this will be pretty easy to calculate
from our original 'cause
if we'd put zero for x,
all we get left is the nine,
and there's our y intercept.
So, sorry about the factoring,
this one I thought it
would factor but it didn't,
but we were able to
use a quadratic formula
and still find our x intercepts.
