Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

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Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 546

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 548

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 546

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 548

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 931

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 931

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

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Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Warning: Cannot modify header information - headers already sent by (output started at /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php:613) in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 1489

Warning: Cannot modify header information - headers already sent by (output started at /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php:613) in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 1489

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(:title Linear Function - Always a line :) >>nav_bar<< %yel%Mathcasts         %exa%Examples      %pra%Practice%%    %mor%More      %up%[[glossary/L]] >><< [[<<]] [table border=1 cellpadding=3 width=100%] [row] []>>&<< %ref b%Definition:%% A %red b%linear function%% is a function whose graph is a line. >>&<<%b reg%Regulations:%% A [[GlossaryT/LinearEquation|linear equation]] in 2 variables is a linear function. >>&<<%b exa%Examples of linear equations:%% %exa%{$ \bbox[border:1px #990000 dotted,1pt]{y=2x-3} $}%% %exa%{$ \bbox[border:1px #990000 dotted,1pt]{2y+5x=2} $}%% %pra%{$ \bbox[border:1px green dotted,1pt]{f(x)=2x-1} $}%% %pra%{$ \bbox[border:1px green dotted,1pt]{3s+2t=2} $}%% >>&<< [tableend] >>-<< [table border=1 cellpadding=3 width=825] [row] [ ] >>&<<%b reg%Regulations:%% There are 2 useful forms of a linear function: %pra%Slope-Intercept Form%% and %ref%Standard Form%% >>&<< [row] []%b ref%Standard Form:%% %pra%{$ Ax+By=C $}%%, where A, B and C are numbers (:showhide init=show div=div22 lshow="+" lhide="-":) More (:div22 id=div22 style="margin-left:20px":) >>&<<%b reg%Regulations:%% The standard form of a line is NOT unique. >>&<>&<<%b reg%Regulations:%% The slope-intercept form of a line is unique. >>&<>&<< [row] [](:showhide init=hide div=div1 lshow="+" lhide="-":) InterActivity (:showhide init=hide div=div3 lshow="+" lhide="-":) %sol s9%Directions for InterActivity%% (:div3 id=div3 class=s9 style="margin-left:20px" :) *Look at the left in the algebra window. There are two lines %ref%a%% and %pra%b%%. Look at the drawing pad. There is only one line. That is because %ref%a%% and %pra%b%% are the same line. %ref%a: {$ x+2y=6 $}%% and %pra%b: {$ y=-0.5x+3 $}%%. On a piece of paper solve %ref%{$ x+2y=6 $}%% for the variable y and see if you get %pra% {$ y=-0.5x+3 $}%% *Multiply %ref%{$ x+2y=6 $}%% by the number 2. What is the result? (:showhide init=hide div=div7111 lshow="Answer" lhide="2x+4y=12":) *Type this result in the input field at the bottom of the interactivity and hit enter. **You should NOT see a different line since this is the same line. **Notice that %b dkr%line c:%% has the same numbers as %b ref%line a%%. GeoGebra has reduced it (divided by 2). *In the input field, type -3x+y=1 and hit enter. You will get a new line. **On a piece of paper, solve this equation for y. >>&<< --> Write your answer in the form y=ax+b. What is your result? (:showhide init=hide div=div7112 lshow="Answer" lhide="y=3x+1":) **In the input field, type in this result and hit enter. You should NOT see a new line. **Think of another way to input this same line. (:showhide init=hide div=div7113 lshow="Sample Answer" lhide="6x-2y=-2":) **Input your result - the lines should coincide AND in the algebra window GeoGebra should reduce your answer so lines d and f have same numbers. (:div3end :) >>&<< (:div1 id=div1 :) >>&<<(:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="810" height="400" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/linearfunc1.ggb" showToolBar="true" showMenuBar="true" showAlgebraInput="true":) (:div1end :) >>&<< >>&<< [tableend] >>-<< ---- >>&<< %rel%Related Topics *[[Lines]] *[[GlossaryT/Slope|Slope of a Line]] *[[GlossaryT/Systems2|Systems of Linear Equations 2x2]] *[[GlossaryT/LinearEquation|Linear Equations]] *[[Cartcoor2|Cartesian Coordinates - Points in the Plane]] >>&<< ---- >>&<< [table width=100%] [row] [][[Glossary/S| Attach:main/tri_purple_up_a.gif ]] [[Glossary/S| Up one level]] [r](:html:) (:htmlend:) [tableend] (:div7111 id=div7111 :) (:div7111end :) (:div7112 id=div7112 :) (:div7112end :) (:div7113 id=div7113 :) (:div7113end :)