Step By Step Calculus » 3.2 - Parabola

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Synopsis
We defined a parabola in geometric terms. In algebraic terms it can be described as the curve traced out by a point that moves in a plane so that its distance from a given point is always equal to its distance from a given line. The given point is called the focus and the given line is called the directrix of the parabola. A parabola with the focus at (p,0) (p,0) and its directrix x=-p x=-p is expressed by the equation

\displaystyle {\textbf{(Equation of Parabola)}\qquad\qquad y^2=4px.}\displaystyle {\textbf{(Equation of Parabola)}\qquad\qquad y^2=4px.}


The Standard Form of a general parabola is given here.
(Standard Form)
\displaystyle { \left\{\begin{array}{ll} (y-k)^2=4p(x-h) & \textrm{ vertical directrix; opens right }(p>0)\textrm{ or left }(p<0)\\ (x-h)^2=4p(y-k) & \textrm{ horizontal directrix; opens up }(p>0)\textrm{ or down }(p<0) \end{array}\right.,}\displaystyle { \left\{\begin{array}{ll} (y-k)^2=4p(x-h) & \textrm{ vertical directrix; opens right }(p>0)\textrm{ or left }(p<0)\\ (x-h)^2=4p(y-k) & \textrm{ horizontal directrix; opens up }(p>0)\textrm{ or down }(p<0) \end{array}\right.,}


where (h,k) (h,k) is the vertex of the parabola. The focus and directrix in the first case are \left(h+p,k\right) \left(h+p,k\right) and x=h-p x=h-p while those for the second case are \left(h,k+p\right) \left(h,k+p\right) and y=k-p y=k-p
In order to sketch a parabola, it is always better to draw the focal chord, the line segment through the points (h+p,k-2p),(h+p,k),(h+p,k+2p)(h+p,k-2p),(h+p,k),(h+p,k+2p) for a left/right parabola or (h-2p,k+p),(h,k+p),(h+2p,k+p)(h-2p,k+p),(h,k+p),(h+2p,k+p) for a up/down parabola.
We prefer to use the following STAPLED procedure when sketching a parabola.
Standard Convert the given equation to standard form.
Type Indicate which way (Up, Down, Left, Right) the parabola opens.
Axes Draw xx and yy axes on a plane.   Use the same scale for both.
Points Mark the focus and vertex on this xyxy-plane.
Lines Draw the directrix and the focal chord.
Ends Make the ends of the focal chord clear.
Draw Draw the parabola: Make sure it goes through the vertex and the ends of the focal chord.   Also, ensure that the distance from each point to the focus is equal to the distance from that point to its projection on the directrix.