Parabola
A parabola is a symmetric "U"-shaped curve or conic
section: a plane curve formed by the intersection of a right circular cone
with a plane parallel to a side of the cone; a 2nd degree or quadratic
function of the form "f(x) = ax2 + bx + c." Formal Definition:
A parabola is the set of all points in a plane that are equidistant from
a given point (called the focus
of the parabola) and a given line (called the directrix
of the parabola).
There are actually two standard forms for the equation
of a parabola. The most common form is is that found in traditional function
format: "f(x) = ax2 + bx + c" or "y = ax2 + bx +
c."
An alternate form is: "(x - h)2 = 4p(y
- k)." This second notation is particularly desirable when one wants sight
recognition of a parabola's vertex, axis of symmetry, the location of the
focus, and the location of the directrix.
In this treatment, only the traditional function format of a parabola
will be discussed ... as shown above. Any function or equation that can
be put into this format is a parabola. The "a" value must be a non-zero
real number; the "b" and "c" values can be any real numbers (& potentially
can be equal to zero - one or both). The basic graph of a parabola is a
symmetric "U"-shaped curve either up or down. If the "a" value is positive,
the curve will be "U"-shape up; if the "a" value is negative the curve
will be "U"-shape down.
A typical parabola is shown in the image below:
In graphing and/or analyzing a parabola, it is normally important to
find the following key elements: whether the parabola is concave up or
down, the vertex of the parabola (which also provides the axis of symmetry
for the "U"-shape), and the x-intercepts of the parabola (if any). In addition,
depending on the situation, it might be desirable to find where the curve
is increasing and where it is decreasing -and- what the range of the function
is.
The x-intercepts: Finding the x-intercepts
of a parabola is a relatively easy task. Basically, the x-intercepts occur
when the "y"-value of the function, when the function itself, is zero.
The x-intercepts, therefore, can be found by equating the function to zero.
In the example (from the diagram above), you would solve the sub-equation:
x2 - 2x - 3 = 0
This can be solved either by factoring or by using the quadratic formula.
In this particular example, the solutions are -1 and 3. These are the x-intercepts
... where the "U"-shape parabola intersects the horizontal x-axis.
Other: If the equation has two solutions (as per above case) ... if "b2
- 4ac" in the quadratic formula is positive, then the parabola intersects
the x-axis at two different distinct points. If the equation has only one
solution (one root repeated twice) ... if "b2 - 4ac" in the
quadratic formula is zero, then the parabola touches the horizontal x-axis
at its vertex point. If the equation has NO real number solutions ... if
"b2 - 4ac" in the quadratic formula is negative, then the parabola
never intersects the horizontal x-axis (the "U" shape is entirely above
or entirely below the x-axis).
The Vertex: The vertex of the parabola is,
so to say, the high or low point on the parabola ... the maximum (max)
or minimum (min) of the parabola; the vertex is that point at the top of
the parabola if it is concave down -or - the point at the bottom of the
parabola if it is concave up. Although there are a variety of techniques
that might be employed to find the vertex, the easiest is to use the front
portion of the quadratic formula: "-b/2a" ... which will provide the x-value
of/for the vertex.
x-valuevertex = -b / 2a
In the example above (from the diagram), the b-value is -2 and the a-value
is 1. The x-value of the vertex is: -(-2) / (2*1) = 1. Once the x-value
of the vertex is found, the y-value of the vertex can be produced by substituting
(plugging in) this x-value into the original function.
y-valuevertex = f( -b/2a )
In the example above (from the diagram), the match y-value of the vertex
is produced by substituting "1" in for "x": 12 - 2 * 1 - 3 =
-4. The full vertex is (1,-4).
Making a coordinate graph ... possibly locating a few additional points
around the vertex (finding y-value match-ups for x-values near the vertex
and them graphing them)... is frequently a reliable method of finding further
information about the parabola.
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and Quadratic Equations
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