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(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.