(:title Quadratic Function:) >>nav_bar<<%reg%Mathcasts         %exa%Examples      %pra%Practice%%    %mor%More      %up%[[glossary/Q]] >><< [[<<]] ---- [table border=1 cellpadding=5 width=98%] [row] []>>&<< %def b%Definition:%% A [[glossaryT/polynomial]] of second degree in one variable is called a %red b%quadratic function%% or %red%quadratic equation%%. >>&<< The general form of a quadratic function is: >>&<< (:div2 style="margin-left:40px":) {$\color{red}{ f(x)=ax^2+bx+c}$}    или    {$\color{red}{y=ax^2+bx+c}$}   where    {$\color{blue}{ a \ne 0}$}. %% >>&<< (:div2end:) ---- >>&<< (:showhide init=hide div=div1 lshow="+" lhide="-":) %exa b%Interactivity%% %sol s9%(:showhide init=hide div=div12 lshow="+" lhide="-":) Directions for interactivity%% (:div1 id=div1 style="margin-left:20px":) >>-<< (:div12 id=div12 :) >>s9<< Click and drag the slider buttons to change the function. >>&<< %pra b%   BTW: %pra%Don't be fooled.%% Changing %yel%c%% does %c11 red%not%% change the size of the quadratic. It only moves it up and down! >><< (:div12end :) >>&<< (:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="800" height="350" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/quadratic_func.ggb" showResetIcon="true" showToolBar="false" showMenuBar="false":) (:div1end :) >>&<< ---- >>&<< %exa b%Example:%%   {$f(x)=3x^2+2x-1$}   is a quadratic function   {$а=3 \quad b=2 \quad c=-1$}    ([[glossary/quadraticFunctionex1|%exa u%other examples]]) >>&<< [table] []%reg b%Rules:%%   * The%pra% graph%% of every quadratic function a %ans%parabola%%. * The intersection of the parabola with {$y$}-axis is unique and is the point (0,с) (see [[y-intercept]]). * The intersections of the parabola with {$x$}-axis (if there is any), are the roots of the function and can be found with the [[quadratic formula]]. (:div align=right:) The graph shown at right is of the quadratic function: %pra s12%{$y=x^2-x-6$}%%   --> (:divend:) [c width=230] {gg width=200; height=200; xmin=-5; xmax=5; xscl=1; ymin=-8; ymax=3; yscl=1; axes(); stroke="#009900"; strokewidth=3; plot("x^2-x-6"); strokewidth=2; stroke="#660000"; dot([-2,0],"closed"); a=text([-2,0],"root",aboveleft); dot([3,0],"closed"); b=text([3,0],"root",aboveright); stroke="#660099"; dot([0,-6],"open"); c=text([0,-6],"y-intercept",belowleft) gg} [tableend] >>&<< ------------- >>&<< %rel%Related topics: * [[quadratic formula]], [[root]] of a function, [[x-intercept]] * [[y-intercept]] * [[polynom]] (polynomial function) ** [[linear function]] >>&<< ---- (:showhide3 init=hide div=box4 lshow='Add comment' lhide='Close':) >>id=box4<< (:commentbox:) >><< [table width=100%] [row] [][[glossary/Q| Attach:main/tri_purple_up_a.gif ]] [[glossary/Q| Up one level]] [r](:html:) (:htmlend:) [tableend]