1. Parabolas

2. Lesson 1
Find the standard form of a quadratic
function, and then find the vertex, line of
symmetry, and maximum or minimum
value for the defined quadratic function.

3. The Parabola is defined as "the set of all points P in a plane equidistant
from a fixed line and a fixed point in the plane." The fixed line is called
the directrix, and the fixed point is called the focus.
A parabola, as shown on the cables of the Golden Gate Bridge
(below), can be seen in many different forms. The path that a thrown
ball takes or the flow of water from a hose each illustrate the shape of
the parabola.

4. Each parabola is, in some form, a graph of a second-degree function and
has many properties that are worthy of examination. Let's begin by
looking at the standard form for the equation of a parabola.
The standard form is (x - h)2 = 4p (y - k), where the focus is (h, k + p) and
the directrix is y = k - p. If the parabola is rotated so that its vertex is
(h,k) and its axis of symmetry is parallel to the x-axis, it has an equation
of (y - k)2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x
= h - p.
It would also be in our best interest to cover another form that the equation
of a parabola may appear as
y = (x - h)2 + k, where h represents the distance that the parabola has
been translated along the x axis, and k represents the distance the
parabola has been shifted up and down the y-axis.

5. Completing the square to get the standard form of a parabola.
We should now determine how we will arrive at an equation in the form y = (x - h)2
+ k;
Example 1
Suppose we are given an equation like
y = 3x2 + 12x + 1.
We now need to complete the square for this equation. I will assume that you have
had some instruction on completing the square; but in case you haven't, I will go
through one example and leave the rest to the reader.
When completing the square, we first have to isolate the Ax2 term and the By term
from the C term. So the first couple of steps will only deal with the first two parts of
the trinomial.
In order to complete the square, the quadratic in the form y = Ax2 + By + C cannot
have an A term that is anything other than 1. In our example, A = 3; so we now
need to divide the 3 out, but that is only out of the 3x2 + 12x terms.
This simplifies to y = 3(x2 + 4x) + 1. From here we need to take 1/2 of our B term,
then square the product. So in this case, we have 1/2(4) = 2, then 22 is 4. Now,
take that 4 and place it inside the parenthetical term.

6. To update what we have: y = 3(x2 + 4x + 4) + 1; but we now need to keep
in mind that we have added a term to our equation that must be
accounted for. By adding 4 to the inside of the parenthesis, we have
done more than just add 4 to the equation. We have now added 4 times
the 3 that is sitting in front of the parenthetical term. So, really we are
adding 12 to the equation, and we must now offset that on the same
side of the equation. We will now offset by subtracting 12 from that 1
we left off to the right hand side.
To update: y = 3(x2 + 4x + 4) + 1 - 12. We have now successfully
completed the square. Now we need to get this into more friendly
terms. The inside of the parenthesis (the completed square) can be
simplified to (x + 2)2. The final version after the smoke clears is y = 3(x
+ 2)2 - 11. And , oh the wealth of information we can pull from
something like this! We will find the specifics from this type of equation
below.

7. Finding the vertex, line of
symmetry, and maximum and
minimum value for the defined
quadratic function.
Let's first focus on the second
form mentioned, y =(x - h)2 +
k. When we have an equation
in this form, we can safely say
that the 'h' represents the
same thing that 'h' represented
in the first standard form that
we mentioned, as does the 'k'.
When we have an equation like
y = (x - 3)2 + 4, we see that the
graph has been shifted 3 units
to the right and 4 units upward.
The picture below shows this
parabola in the first quadrant.

8. Had the inside of the parenthesis in the example equation read,"(x+3)" as
opposed to "(x-3)," then the graph would have been shifted three units to the left
of the origin. The "+4" at the end of the equation tells the graph to shift up four
units. Likewise, had the equation read "-4," then the graph would still be pointed
upward, but the vertex would have been four units below the x-axis.
A great deal can be determined by an equation in this form.

9. The Vertex
The most obvious thing that we can tell, without having to look at the graph, is
the origin. The origin can be found by pairing the h value with the k value, to
give the coordinate (h, k). The most obvious mistake that can arise from this is
by taking the wrong sign of the 'h.' In our example equation, y = (x - 3)2 + 4,
we noticed that the 'h' is 3, but it is often mistaken that the x-coordinate of our
vertex is -3; this is not the case because our standard form for the equation is
y = (x - h)2 + k, implying that the we need to change the sign of what is inside
the parenthesis.

10. The Line of Symmetry
To find the line of symmetry of a parabola in this form, we need to remember
that we are only dealing with parabolas that are pointed up or down in
nature. With this in mind, the line of symmetry (also known as the axis of
symmetry) is the line that splits the parabola into two separate branches that
mirror each other. The line of symmetry goes through the vertex, and since
we are now only dealing with parabolas that go up and down, the line of
symmetry must be a vertical line that will begin with "x = _ ". The number
that goes in this blank will be the x-coordinate of the vertex. For example,
when we looked at y = (x - 3)2 + 4, the x-coordinate of the vertex is going be
3; so the equation for the line of symmetry is x = 3.

11. In order to visualize the line of symmetry, take the picture of the parabola
above and draw an imaginary vertical line through the vertex. If you were to
take the equation of that vertical line, you would notice that the line is going
through the x-axis at x = 3. An easy mistake that students often make is that
they say that the line of symmetry is y = 3 since the line is vertical. We must
keep in mind that the equations for vertical and horizontal lines are the reverse
of what you expect them to be. We always say that vertical means "up and
down; so the equation of the line (being parallel to the y-axis) begins with 'y
=__'," but we forget that the key is which axis the line goes through. So since
the line goes through the x-axis, the equation for this vertical line must be x =
__.

12. The Maximum or Minimum
In the line of symmetry discussion, we dealt with the x-coordinate of the
vertex; and just like clockwork, we need to now examine the y-coordinate.
The y-coordinate of the vertex tells us how high or how low the parabola
sits.
Once again with our trusty example, y = (x-3)2 + 4, we see that the y-
coordinate of the vertex (as derived from the number on the far right of the
equation) dictates how high or low on the coordinate plane that the
parabola sits. This parabola is resting on the line y = 4 (see line of
symmetry for why the equation is y = __, instead of x = __ ). Once we
have identified what the y-coordinate is, the last question we have is
whether this number represents a maximum or minimum. We call this
number a maximum if the parabola is facing downward (the vertex
represents the highest point on the parabola), and we can call it a
minimum if the parabola is facing upward (the vertex represents the
lowest point on the parabola).

13. How do we tell if the parabola is
pointed upward or downward by
just looking at the equation?
As long as we have the equation in the form derived from the completing the
square step, we look and see if there is a negative sign in front of the
parenthetical term. If the equation comes in the form of y = - (x - h)2 + k, the
negative in front of the parenthesis tells us that the parabola is pointed
downward (as illustrated in the picture below). If there is no negative sign in
front, then the parabola faces upward.