document.write( "Question 1208044: Given f(x) = -4x + 1,\r
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Algebra.Com's Answer #846184 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
Given f(x) = -4x + 1,\r
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As x goes from 2 to 5, y [or f(x)] goes from f(2)=-4(2)+1=-7 to f(5)=-4(5)+1=-19\r\n" );
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document.write( "From -7 to -19 is a change of (-19)-(-7)=-19+7=-12 in y\r\n" );
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document.write( "From 2 to 5 is a change of 5-2 = +3 in x\r\n" );
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document.write( "So the average change in y [or f(x)] per the average change in x\r\n" );
document.write( "is -12 units change in y per a +3 change in x, which means an average\r\n" );
document.write( "change of (-12)/(+3)=-4 y-units change per x-units change.\r\n" );
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document.write( "That's what the slope is, the change in y, divided by the change in x \r\n" );
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document.write( "Answer: -4 change in f(x) y-units per x-unit change.

B. Find an equation of the secant line containing (2, f(2)) and (5, f(5)).

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document.write( "change y to f(x)\r\n" );
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document.write( "So the secant line coincides with the graph of f(x).\r\n" );
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document.write( "[That happens when the graph of the given function \r\n" );
document.write( "f(x), is a line, but not when the given function curves].\r\n" );
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document.write( "Here is a case when the given function f(x) curves, and\r\n" );
document.write( "the secant line does not coincide with the given function f(x)\r\n" );
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document.write( "The black line is the secant line. It only has two points\r\n" );
document.write( "in common with the function f(x) (the red curve).\r\n" );
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document.write( "But your problem here is not a curve: \r\n" );
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document.write( "The case you were given was a special case when the \r\n" );
document.write( "secant line happens to coincide with the function f(x),\r\n" );
document.write( "like what you see below, where the black line, which is \r\n" );
document.write( "the secant line, coincides with the red line, which is the \r\n" );
document.write( "graph of f(x).\r\n" );
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document.write( "Your teacher gave you a very special case, but most graphs\r\n" );
document.write( "for f(x) are not straight lines, and the secant lines\r\n" );
document.write( "will usually have only two points in common with the graph\r\n" );
document.write( "of the given function f(x) like the graph above.\r\n" );
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document.write( "Remember what the word \"secant\" means, a line that cuts\r\n" );
document.write( "through a curve, as you see in the first curve.  \r\n" );
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document.write( "[Remember from your geometry that a secant line was a line\r\n" );
document.write( "that cuts through a circle.] \r\n" );
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document.write( "Your problem here is a special case when the secant line\r\n" );
document.write( "doesn't \"cut through\" the curve of f(x) but coincides with it. \r\n" );
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document.write( "This type problem is to prepare you for calculus, where\r\n" );
document.write( "you will learn about the secant line approaching a tangent\r\n" );
document.write( "line.  You will also remember from your geometry that a\r\n" );
document.write( "tangent line just touches a circle in one point, and does\r\n" );
document.write( "not cut through the circle.  I didn't want you to get\r\n" );
document.write( "confused because your teacher gave you such a special case.\r\n" );
document.write( "Just remember that most of your problems will be like the\r\n" );
document.write( "curve in the first graph above, not the lower special case,\r\n" );
document.write( "which you were given.\r\n" );
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document.write( "Edwin

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