Difference between revisions of "Parabola"
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=== Olympiad === | === Olympiad === | ||
+ | Find the area of the largest triangle <math>ABC</math> [and prove this is the maximum] whose interior is entirely within the region bounded by <math>y=\sqrt{3}x-1</math> and <math>y=3x^2-12x+1</math>. | ||
== See also== | == See also== |
Revision as of 12:17, 30 May 2011
A parabola is a type of conic section. A parabola is a locus of points that are equidistant from a point (the focus) and a line (the directrix).
Contents
Parabola Equations
There are several "standard" ways to write the equation of a parabola. The first is polynomial form: where a, b, and c are constants. This is useful for manipulating the polynomial.
The second is completed square form, or where a, h, and k are constants and the vertex is (h,k). This is very useful for graphing the quadratic because the vertex and stretching factor are immediately before you.
The third way is the conic section form, or or
where the p is a constant, and is the distance from the focus to the vertex.
Graphing Parabolas
Using the completed square form, or
, the vertex of the graph is at the point
. The graph appears vertically if the
term is squared, and horizontal if the
term is squared. The graph will be oriented (opens up) upwards/right if
is positive, and will be downwards/left if
is negative.
Problems
Introductory
- A parabola with equation
passes through the points (2,3) and (4,3). What is
?
Intermediate
Suppose that a parabola has vertex and equation
, where
and
is an integer. The minimum possible value of
can be written in the form
where
and
are relatively prime positive integers. Find
.
Olympiad
Find the area of the largest triangle [and prove this is the maximum] whose interior is entirely within the region bounded by
and
.