excelmath Mrs. Twu
University of Toledo, Excel Progeam  
 
Linear, Quadratic, and Cubic Functions

Linear, Quadratic, and Cubic Functions


These are names for functions of first, second and third order polynomial functions, respectively.  What this means is that the highest order of x (the variable) in the function is 1, 2 or 3.

The generalized form for a linear function (1 is highest power):
f(x) = ax+b,  where a and b are constants, and a is not equal to 0

The generalized form for a quadratic function (2 is highest power):
f(x) = ax2+bx+c,  where a, b and c are constants, and a is not equal to 0

The generalized form for a cubic function (3 is highest power):
f(x) = ax3+bx2+cx+d,  where a, b, c and d are constants, and a is not equal to 0

The roots of a function are defined as the points where the function f(x)=0.  For linear and quadratic functions, this is fairly straight-forward, but the formula for a cubic is quite complicated and higher powers get even more involved.  I will go over the derivation of the first two now, But the rest plus a history of the discoveries can be found here.

A linear equation is very simple to solve for f(x)=0:

0   = ax+b
-ax = b
 x  = b/(-a) = -b/a
,  a not equal to 0

The equation for the root of a quadratic is only slightly more complex.  The idea is to isolate x by putting the left side into the form (x+q)2 and then taking the square root.  We do this by some nifty algebra:
ax2 + bx + c = 0
x2 + (b/a)x + c/a = 0
[x2 + (b/a)x + b2/4a2] - b2/4a2 + c/a = 0
[x + (b/2a)]2 - ( b2/4a2 - c/a) = 0
[x + (b/2a)]2 =  b2/4a2 - c/a
x+(b/2a) = (+/-) sqrt(b2/4a2-4ac/4a2)
x = -b/2a (+/-) sqrt(b2-4ac)/2a
x+(b/2a) = (+/-) sqrt(b2/4a2-4ac/4a2)
x = (-b (+/-) sqrt(b2-4ac))/2a
Try to get (x+g)2 = x2 + (b/a)x + ??
(x+(1/2)(b/a))2 = x2 + 2(1/2)(b/a)x + (1/4)(b2/a2)
(x+b/2a)2 = x2 + (b/a)x + b2/4a2

-David Eger


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Last updated  2008/09/28 06:29:42 PDTHits  154