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% Document Info
\author{starlight (starlighttx)}
\pdfinfo{
  /Title (la-parabola.pdf)
  /Creator (Cheatography)
  /Author (starlight (starlighttx))
  /Subject (La Parabola Cheat Sheet)
}

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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{La Parabola Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{starlight (starlighttx)} via \textcolor{DarkBackground}{\uline{cheatography.com/211272/cs/45738/}}}
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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}starlight (starlighttx) \\
  \uline{cheatography.com/starlighttx} \\
  \end{tabulary}
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\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 23rd February, 2025.\\
   Updated 23rd February, 2025.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.74195 cm} x{3.23505 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{{\bf{La parabola}}}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{La parabola è una delle coniche. Fissati nel piano un punto {\emph{F}} (fuoco) e una retta {\emph{d}} (direttrice), si chiama parabola la curva luogo geometrico dei punti equidistanti da {\emph{F}} e da {\emph{d}}} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
EQUAZIONE & y= ax\textasciicircum{}2\textasciicircum{} + bx + c \tn 
% Row Count 5 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.43873 cm} x{2.53827 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{{\bf{Formule generali: parallela all'asse y}}}}  \tn
% Row 0
\SetRowColor{LightBackground}
VERTICE & V(-b/2a ; -Δ/4a) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
FUOCO & F(-b/2a ; 1-Δ/4a) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
DIRETTRICE & y=-(1+Δ/4a) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
ASSE DI SIMMETRIA & x=-b/2a \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.43873 cm} x{2.53827 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{{\bf{Formule generali: perpendicolare all'asse y}}}}  \tn
% Row 0
\SetRowColor{LightBackground}
EQUAZIONE & x= ay\textasciicircum{}2\textasciicircum{} +by +c \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
VERTICE & V(-Δ/4a ; -b/2a) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
FUOCO & F(1-Δ/4a ; -b/2a) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
DIRETTRICE & x=-(1+Δ/4a) \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
ASSE DI SIMMETRIA & y= -b/2a \tn 
% Row Count 5 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{3.68298 cm} x{1.29402 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Parabole congruenti}}  \tn
% Row 0
\SetRowColor{LightBackground}
Date y=ax\textasciicircum{}2\textasciicircum{}+bx+c e y=a\textasciitilde{}1\textasciitilde{}x\textasciicircum{}2\textasciicircum{}+b\textasciitilde{}1\textasciitilde{}x+c\textasciitilde{}1\textasciitilde{}, esse sono congruenti se hanno la stessa apertura & {\bf{|a|=|a\textasciitilde{}1\textasciitilde{}|}} \tn 
% Row Count 4 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{SEGMENTO PARABOLICO}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/starlighttx_1740320756_segmento parabolico.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.83689 cm} x{2.14011 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Segmento parabolico}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Il segmento parabolico è la parte di piano compresa tra la parabola e la retta secante} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Per il teorema di Archimede, l'area del segmento parabolico è i 4/3 di quella del triangolo massimo inscritto al segmento parabolico che ha base AB & A\textasciitilde{}sp\textasciitilde{}=A\textasciitilde{}tm\textasciitilde{}(4/3) \tn 
% Row Count 9 (+ 7)
% Row 2
\SetRowColor{LightBackground}
Formula generale & A\textasciitilde{}sp\textasciitilde{}= |a|/6 × |x\textasciitilde{}0\textasciitilde{}-x\textasciitilde{}1\textasciitilde{}|\textasciicircum{}3\textasciicircum{} \tn 
% Row Count 11 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{RETTA ESTERNA ALLA PARABOLA}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/starlighttx_1740319553_EST.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Retta esterna alla parabola}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{La retta è esterna alla parabola quando non hanno punti in comune} \tn 
% Row Count 2 (+ 2)
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\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Δ \textless{} 0} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{RETTA SECANTE ALLA PARABOLA}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/starlighttx_1740319646_SEC.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Retta secante alla parabola}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{La retta e la parabola hanno due punti di intersezione distinti} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Δ\textgreater{}0} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{RETTA TANGENTE ALLA PARABOLA}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/starlighttx_1740319810_TAN.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Retta tangente alla parabola}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{La retta e la parabola hanno un punto di intersezione doppio (= due coincidenti)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Δ=0} \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Punti e parabole}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sistema di equazioni tra fascio di rette e parabola & y-y\textasciitilde{}0\textasciitilde{}=m(x-x\textasciitilde{}0\textasciitilde{})    y=ax\textasciicircum{}2\textasciicircum{}+bx+c \tn 
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% Row 1
\SetRowColor{white}
Equazione risolvente del sistema di equazioni tra fascio di rette e parabola & ax\textasciicircum{}2\textasciicircum{}+bx+c=m(x-x\textasciitilde{}0\textasciitilde{})+y\textasciitilde{}0\textasciitilde{} \tn 
% Row Count 7 (+ 4)
% Row 2
\SetRowColor{LightBackground}
Se il punto è esterno alla parabola, possiamo tracciare due tangenti & con Δ=0, abbiamo soluzioni m\textasciitilde{}1\textasciitilde{} diverso da m\textasciitilde{}2\textasciitilde{} \tn 
% Row Count 11 (+ 4)
% Row 3
\SetRowColor{white}
Se il punto appartiene alla parabola, possiamo tracciare una sola tangente & con Δ=0, abbiamo una sola soluzione di m \tn 
% Row Count 15 (+ 4)
% Row 4
\SetRowColor{LightBackground}
Se il punto è interno alla parabola, non possiamo tracciare nessuna tangente & con Δ=0, non abbiamo alcuna soluzione \tn 
% Row Count 19 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{SE IL PUNTO APPARTIENE ALLA PARABOLA}}  \tn
% Row 0
\SetRowColor{LightBackground}
Formula per trovare m (solo nella parabola) & m=2ax\textasciitilde{}0\textasciitilde{}+b, avendo P(x\textasciitilde{}0\textasciitilde{};y\textasciitilde{}0\textasciitilde{}) \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Formule generali (ogni curva) & x-\textgreater{}(x+x\textasciitilde{}0\textasciitilde{})/2;  y-\textgreater{}(y+y\textasciitilde{}0\textasciitilde{})/2;  x\textasciicircum{}2\textasciicircum{}-\textgreater{}x×x\textasciitilde{}0\textasciitilde{} \tn 
% Row Count 6 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{FASCI DI PARABOLE}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/starlighttx_1740321195_fasci.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.04057 cm} x{2.93643 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Fasci di parabole}}  \tn
% Row 0
\SetRowColor{LightBackground}
Equazione & y=ax\textasciicircum{}2\textasciicircum{}+bx+c + k(a\textasciitilde{}1\textasciitilde{}x\textasciicircum{}2\textasciicircum{}+b\textasciitilde{}1\textasciitilde{}x+c) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Punti base & sono i punti in comune tra tutte le parabole del fascio \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Parabole degeneri algebriche & sono rette passanti per i punti base che si possono ottenere dall'equazione del fascio \tn 
% Row Count 9 (+ 4)
% Row 3
\SetRowColor{white}
Parabole degeneri grafiche & sono rette/coppie di rette parallele all'asse y passanti per i punti base. Le possiamo identificare nel grafico, ma non si possono ricavare dall'equazione \tn 
% Row Count 16 (+ 7)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Fasci di parabole: casi grafici}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{CASO 1: due punti base e una parabola degenere algebrica (+ una coppia di rette passante per punti base -\textgreater{} parabola degenere grafica)} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{CASO 2: un punto base doppio e una parabola degenere algebrica (+ una retta passante per il punto base -\textgreater{} parabola degenere grafica)} \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{CASO 3: nessun punto base, ma una parabola degenere algebrica} \tn 
% Row Count 8 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{CASO 4: nessun punto base, nessuna degenere, ma congruenti e con stesso asse di simmetria} \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{CASO 5: un punto base semplice, senza degenere algebrica (+ una una retta passante per il punto base -\textgreater{} parabola degenere grafica)} \tn 
% Row Count 13 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{3.08574 cm} x{1.89126 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Fasci di parabole: casi algebrici}}  \tn
% Row 0
\SetRowColor{LightBackground}
CASO 1: la y NON dipende dal parametro & es. y=3tx\textasciicircum{}2\textasciicircum{}-x+t \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
CASO 2: anche la y dipende dal parametro & es. {\bf{t}}y=(t-1)x\textasciicircum{}2\textasciicircum{}+tx-2 \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{LA NORMALE}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/starlighttx_1740322532_normale.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{La normale}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{La normale è la retta passante per il punto di tangenza che è {\bf{perpendicolare}} alla tangente stessa} \tn 
% Row Count 3 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}