\r\n" );
document.write( "We will change the slope first and then we will slide it down\r\n" );
document.write( "\r\n" );
document.write( "In the equation:\r\n" );
document.write( "\r\n" );
document.write( "y = 
x + 4, is of the form y = mx+b\r\n" );
document.write( "\r\n" );
document.write( "The 4 tells us where the line crosses the y-axis.\r\n" );
document.write( "\r\n" );
document.write( "the tells us how steep the line is. The numerator\r\n" );
document.write( "of is 1 and the denominator is 3. That tells\r\n" );
document.write( "us that the line goes up 1 unit for every 3 units it goes\r\n" );
document.write( "to the right. \r\n" );
document.write( "\r\n" );
document.write( "So it looks like this:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "You will notice in the right graph that the green lines show that it \r\n" );
document.write( "rises 1 unit for every 3 units it moves to the right. \r\n" );
document.write( "\r\n" );
document.write( "Now we want to change the slope by a factor of 3. That is, we\r\n" );
document.write( "want to make the line 3 times as steep. So that instead of rising\r\n" );
document.write( "only 1 unit for every 3 units it runs, we want it to rise 3 times\r\n" );
document.write( "as many units for every 3 units it moves to the right. To do that\r\n" );
document.write( "we will multiply the slope by 
making it 
\r\n" );
document.write( "or 
. Yes I know that reduces to 1 but I am leaving the \r\n" );
document.write( "denominator as 3, to show that it is now 3 times as steep as the \r\n" );
document.write( "original line. I haven't changed the 4 yet. I will do that next\r\n" );
document.write( "when I slide it down 5 units. Here is the graph of \r\n" );
document.write( "\r\n" );
document.write( "y = x + 4\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "And again the green lines on the right show that this new line rises\r\n" );
document.write( "3 times as much per 3 units of movement to the right as the original\r\n" );
document.write( "line did. And now I will reduce the to just 1 and the\r\n" );
document.write( "equation of this new line is\r\n" );
document.write( "\r\n" );
document.write( "y = 1x + 4 \r\n" );
document.write( "\r\n" );
document.write( "or just \r\n" );
document.write( "\r\n" );
document.write( "y = x + 4\r\n" );
document.write( "\r\n" );
document.write( "Now since the 4 tells us where the line crosses the y-axis, and we want\r\n" );
document.write( "the line to slide down the y-axis by 5 units, we simply subtract 5\r\n" );
document.write( "from the 4 getting -1. So this is the final graph of\r\n" );
document.write( "\r\n" );
document.write( "y = x - 1\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "In other word, take the equation\r\n" );
document.write( "\r\n" );
document.write( "y = x + 4, multiply the slope by 3 and subtract 5\r\n" );
document.write( "from the 4:\r\n" );
document.write( "\r\n" );
document.write( "y = 3·x + 4 - 5\r\n" );
document.write( "\r\n" );
document.write( "y = x - 1\r\n" );
document.write( "\r\n" );
document.write( "So you might ask why I didn't just say \"multiply by 3 \r\n" );
document.write( "and subtract 5 from 4 in the beginning.\r\n" );
document.write( "If so my answer is: because it is extremely important that you\r\n" );
document.write( "understand what you are doing GRAPHICALLY! It's not enough\r\n" );
document.write( "just to say \"multiply and subtract\". What makes math hard is\r\n" );
document.write( "trying to memorize what to do and ignoring why you do it. Once\r\n" );
document.write( "you understand why you do what you do in math, math becomes\r\n" );
document.write( "easy. Memorizing a thousand rules of \"what to do\" without\r\n" );
document.write( "understanding why you do them is almost impossible.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
