Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lectures, we introduced "quadratic functions".
In this lecture, we will examine two standard ways to write a quadratic function
the general form
and the vertex form
and see how those forms are related to the function's graph.
We have seen that any quadratic function of x
can be written as "a x squared, plus b x, plus c"
where a, b, and c are constants that determine the shape and position of the function's graph.
We will refer to this form of a quadratic function as the "general form".
So let's see how in general form, the constants a, b, and c affect the function's graph.
As we have seen, the graph of any single-variable quadratic function
always has the same basic shape called a "parabola".
In general form, the constants b and c do not affect the parabola's shape
only its position.
When b and c are zero
the parabola's vertex will lie at the origin.
Only the constant "a" affects the parabola's shape
determining whether the parabola opens up or down, and how tightly it curves.
Increasing the value of "a" stretches the parabola vertically
producing a narrower parabola.
and decreasing the value of "a" shrinks the parabola vertically
producing a wider parabola.
As the value of "a" approaches zero, the parabola approaches a horizontal line
but since the value of "a" in a quadratic function cannot be zero
the parabola must always have some curvature.
Negative values of "a" will cause the parabola to open down
and just as with positive values of "a"
negative values of larger magnitude will produce narrower parabolas.
The constants b and c don't affect the shape of the parabola
but together, a, b, and c all affect the position of its vertex.
When b and c are both zero the parabola's vertex will lie at the origin.
Since c adds a constant value to the function at every point
changing c causes the parabola to shift vertically
either up when c is positive
or down when c is negative.
Changing the value of b also causes the parabola's position to shift
but in a more complicated way.
Positive values of b cause the parabola to shift down and to the left
and negative values cause the parabola to shift down and to the right.
So by adjusting the values of both b and c
the parabola, whose shape is defined by "a"
can be positioned at any point on the Cartesian plane.
However, it is not always intuitively obvious which values to choose for b and c
in order to position the parabola at a given point.
To make the position of a parabola more obvious from its mathematical form
quadratic functions are often written in what is called "standard" or "vertex form".
In both the general and vertex forms
the constant "a" determines how tightly the parabola curves
and whether it opens up or down.
However, the constants b and c in the general form
and h and k in the vertex form
affect the parabola's position in different ways.
Unlike b and c in the general form
in the vertex form, h and k are the actual horizontal and vertical coordinates of the vertex.
This allows us to identify the coordinates of a parabola's vertex
simply by looking at the function.
For example, the horizontal and vertical coordinates of this parabola's vertex are 4
and 2.
This makes it much easier to sketch the graph of a quadratic function written in vertex form
since once the direction that a parabola opens and the position of its vertex are known
parabolas have the same basic shape.
The only difference, which is determined by the magnitude of "a"
is that the parabola may be wider
or narrower.
For example, let's say that we wish to sketch the graph of the quadratic function
"one-third, x minus three squared, minus one".
In this function, the values of h and k are three and negative one
which tells us the coordinates of the parabola's vertex.
And since the sign of "a" is positive, we know that the parabola opens up.
Since all parabola's have the same basic shape, varying only in their width
given just one additional point on the parabola
we can determine how much the parabola must be stretched horizontally.
To do this, we can pick any convenient value of x
and then solve for the value of f(x) at that point.
For instance, in this example we could pick an x value of zero
and then solve for f(x) to see where the parabola intersects the vertical axis.
Since when x is zero, f(x) is 2
we know that the parabola must be stretched horizontally
until it intersects the vertical axis at 2.
It is obvious that when sketching the graph of a quadratic function written in vertex form
the fact that the constants h and k are the coordinates of the parabola's vertex is helpful.
However, it is not so obvious why h and k are the vertex's horizontal and vertical coordinates.
In the next lecture, we will see why this is the case.