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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Parabolas">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Mar 26 21:56:45 1997}
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\section{Parabolas\label{Parabolas}}

The graph of the equation $y=ax^{2}+bx+c$ is a U-shaped curve called a \emph{%
parabola} that opens either upward or downward, depending on whether the
sign of $a$ is positive or negative. The lowest or highest point of the
parabola is called the \emph{vertex} and the parabola is symmetric about its 
\emph{axis}.

\subsection{Geometric Definition of a Parabola\label%
{Parabola, geometric definition}}

A \textbf{parabola} is the set of points in the place equidistant from a
fixed point $F$ (called the \textbf{focus}) and a fixed line $I$ (called the 
\textbf{directrix}).

\subsection{\label{Equation of a parabola with vertex at origin}Parabola
with Vertical Axis}

The graph of the equation 
\[
x^{2}=4py 
\]
is a parabola with the following properties. 
\[
\text{%
\begin{tabular}{ll}
\textbf{Vertex} & $V(0,0)$ \\ 
\textbf{Focus} & $F(0,p)$ \\ 
\textbf{Directrix} & $y=-p$%
\end{tabular}
} 
\]

\begin{enumerate}
\item  The parabola opens upward if $p>0$.\FRAME{dtbpF}{138.0625pt}{94.4375pt%
}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
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\item  The parabola opens downward if $p<0$.\FRAME{dtbpF}{146.1875pt}{%
100.875pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
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\end{enumerate}

\subsection{\protect\vspace{1pt}Parabola with Horizontal Axis}

The graph of the equation 
\[
y^{2}=4px 
\]
is a parabola with the following properties. 
\[
\begin{tabular}{ll}
\textbf{Vertex} & $V(0,0)$ \\ 
\textbf{Focus} & $F(p,0)$ \\ 
\textbf{Directrix} & $x=-p$%
\end{tabular}
\]

\begin{enumerate}
\item  The parabola opens to the right if $p>0$.\FRAME{dtbpF}{143.125pt}{%
101.75pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
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\item  The parabola opens to the left if $p<0$.\FRAME{dtbpF}{144.375pt}{%
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\end{enumerate}

\ 

\subsection{Related Scientific Notebook Topics}

\hyperref{Algebra}{}{}{../../Help/DM3.tex}

\hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}

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