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(:title Root of a function:) >>nav_bar<<%reg%Mathcasts         %exa%Examples      %pra%Practice%%    %mor%More      %up%[[glossary/R]] >><< [[<<]] ---- [table border=1 cellpadding=5 width=98%] [row] []>>&<< %def b%Definition:%% A %red b%root%% of a function is an x-value at which the y-value is zero. >>-<< %reg b%Rules:%% *Let {$ y=f(x) $} be a function. Then {$ x=a $} is a root if {$ f(a)=0 $}. *If {$ x=a $} is a root of {$ f(x) $}, then the graph of {$ f(x) $} crosses the x-axis at a. *%c11%So a root is an %c11 red%[[x-intercept]]. *The x-axis is also the line y=0. So substituting y=0 and solving for x gives roots. >>&<< [table] [row] [] >>&<< %exa b%Example:%% Find the roots of the quadratic function {$ f(x)= x^2-x-2 $}. (:div style="margin-left:20px":)To find the roots of a quadratic function, we use the [[quadratic formula]]. The roots are: %ref%{$ x_1=2 $}%%    and   %mor%{$ x_2=-1 $}%% >>&<>le20 a6<< [table border=2 bordercolor=#ddddff cellspacing=0 cellpadding=5 bgcolor=#eeeeff style='color:#000066; font-size:1em;'] [row] [c width=170] %exa c11% {$ y=-x+2 $}%% [c width=170] %pra c11% {$ y=-x^2+2x-1 $}%% [c width=170] %mor c11% {$ y=2^x $}%% [c width=230] %bgr c11% {$ f(x)=sinx $} [row] [c] %exa% 1 root [c] %pra% 1 roots [c] %mor% 0 roots [c] %bgr% infinitely many roots [row] [c] %exa% root: x=2 [c] %pra% root: x=1 [c] %mor% - [c] %bgr% roots: {$ \left\{ {x|x = k\pi ,\,\,k \in \bf{Z}} \right\} $} [row] [c] {gg width=160; height=200; xmin=-4; xmax=4; xscl=1; axes(); stroke="#660000"; strokewidth=2; plot("-x+2"); dot([2,0],"open"); c=text([2,0],"root",aboveright);gg} [c] {gg width=160; height=200; xmin=-4; xmax=4; xscl=1; axes(); stroke="#006600"; strokewidth=2; plot("-x^2+2x-1"); dot([1,0],"open"); c=text([1,0],"root",aboveright);gg} [c] {gg width=160; height=200; xmin=-4; xmax=4; xscl=1; axes(); stroke="#660099"; strokewidth=2; plot("2^x"); c=text([1.5,1.5],"no roots",right); gg} [c] {gg width=220; height=200; xmin=-4; xmax=7; xscl=1; axes(); stroke="#009999"; strokewidth=2; plot("sin(x)"); dot([-3.14,0],"open"); c=text([-3.14,0],"root",aboveright); dot([0,0],"open"); c=text([0,0],"root",aboveleft);dot([3.14,0],"open"); c=text([6.28,0],"root",aboveleft);dot([6.28,0],"open"); c=text([3.14,0],"root",aboveright);gg} [row] [c width=180] %exa c11% [[linear function]] [c width=180] %pra c11% [[quadratic function]] [c width=180] %mor c11% [[exponential function]] [c width=180] %bgr c11% [[trigonometric function]] [tableend] [tableend] >>&<< ------------- >>&<< %rel%Related topics: * [[x-intercept]] of the graph of a function * [[function]] * [[y-intercept]] of the graph of a function * [[polinom]], [[linear function]], [[quadratic function]] >>&<< ---- (:showhide3 init=hide div=box4 lshow='Add comment' lhide='Close':) >>id=box4<< (:commentbox:) >><< [table width=100%] [row] [][[glossary/R| Attach:main/tri_purple_up_a.gif ]] [[glossary/R| Up one level]] [r](:html:) (:htmlend:) [tableend]