(male narrator)
In this video,
we'll look at graphing
quadratic functions
by identifying
key points.
A quadratic equation
is one that is in the form:
ax squared,
plus bx, plus c.
Any equation that's in this form
will generate this u-shape,
which we call
a "parabola."
The u-shape has several
key pieces of information
that we can pull off
of the equation
to make graphing
it easier.
The first thing
we're interested in
is the direction
of the graph.
The direction of the graph
is based on the first value--a.
If it is greater than 0--
or positive--
the parabola
will open up.
If it less than 0--
or negative--
the parabola
will open down.
This is easy
to remember,
because if a is positive,
the parabola is happy,
and if a is negative,
the parabola is sad.
Another key piece
of information we'll look for
is where the graph
crosses the y-axis.
We call this
the "y-intercept."
The y-intercept
is easy to identify,
because we know the x-value
at this point is 0.
If each of the x's
are equal to 0,
the only thing left
is c.
c will be
the y-intercept.
Similarly, we'll
also be interested
in when the graph crosses
the x-axis at the x-intercepts.
There will always be
two x-intercepts,
as long as
they're real.
And we know
the y-value there is 0,
so we will set
the equation equal to 0
in order to find
the x-intercept.
Zero equals ax squared,
plus bx, plus c,
is quickly solved
by either factoring,
completing the square,
or the quadratic formula.
The last key point we find
is the point where it changes
direction at the bottom or top--
called the "vertex."
The vertex has two components
we need to find.
The first is the x-coordinate,
which we find
by taking the values
for a and b in the equation
using the simple formula:
the opposite of b over 2a.
Notice, this is
the quadratic formula
without the square root.
To find the y-value
of the vertex,
we'll simply plug x
into this function
and evaluate
to see what we get for y.
Let's take a look
at an example
where we identify
this key information
to help us graph
the function.
In this problem, we can start
by identifying the direction
based on the value for a,
which is a +1.
Because that's positive,
we know the direction and shape
will be a happy parabola
opening up.
Next, we can identify
the y-intercept.
If x is 0, the only thing left
would be the -3.
This means the graph
crosses the y-axis at -3.
We have the first point
on our graph.
We can also find the x-intercept
by making the equation equal 0.
When we say x squared,
minus 2x, minus 3, equals 0,
we can quickly solve
by factoring to x, minus 3,
times x, plus 1, equals 0.
Setting each factor
equal to 0,
we can quickly find
the two x-intercepts.
By adding 3,
the first x-intercept is at 3,
on the x-axis;
and subtracting 1
to get our second x-intercept
of -1, on the x-axis.
The only thing left to find
is the vertex,
which we do
by using the formula:
the opposite
of b over 2a.
Remember
from the quadratic formula,
we get a, b, and c
from our coefficients.
So the opposite of b
will be -2, over 2a, over 1.
This reduces to 1,
so 1 must be
the x-value of the vertex.
To find our y,
we plug 1 in,
getting 1 squared, minus 2,
times 1, minus 3;
and evaluate to get y,
equals 1, minus 2, minus 3,
or y equals -4.
The vertex then is
at an xy point: 1,-4.
Plotting this point
and connecting the dots,
we get the u-shape
we would expect,
opening up
in our parabola.
Part 2, 
we'll see another.
