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Determining Linear Relations


Date: 04/22/98 at 10:48:17
From: Missy
Subject: Linear relations

The question states:

Consider the following tables

   A           B            C            D           E
 x    y   |  x    y  |   x      y  |  x     y  |  x     y 
-3   1.25 | -3   -3  |  -3    -32  | -2     7  | -2     2
 0    10  |  0   -1  |   0     -5  |  1     1  |  1     2
 1    20  |  1  -1/3 |   1     -4  |  3    -3  |  3    12
 3    80  |  3    1  |   3     22  |  5    -7  |  5    30

a. Decide which of the table above define linear relations. For 
   each linear relation, find the slope and the y intercept of the 
   line. Write the rules that were used to generate the table.

b. For each non-linear example, find the rule that was used.

Once I get the formula, I can figure out the answer, but I can not 
figure out what the formula is.


Date: 04/22/98 at 15:06:51
From: Doctor Rob
Subject: Re: Linear relations

To determine if there is or isn't a linear relation, pick the first 
two points (x1,y1) and (x2,y2) and compute the slope of the line 
connecting them, m12 = (y2-y1)/(x2-x1). Do this for the other pairs of 
adjacent points from the same table, computing m23 and m34. If all 
these values are the same, the relation is linear, and the slope 
is that common value m. Then use the point-slope form of the equation 
of a line: y = m*x + (y1-m*x1), and read off the y-intercept 
y1 - m*x1. If the slopes are not the same, there is no linear 
relation.

To find a quadratic relation, y = a*x^2 + b*x + c, compute m12, 
m23, and m34 as above, and then compute M13 = (m23-m12)/(x3-x1) and
M24 = (m34-m23)/(x4-x2). If these are equal, they are the value of a.
Then b = m12 - a*(x1+x2), and c = y1 - a*x1^2 - b*x1. If M13 and M24 
are not equal, there is no quadratic relation.

To find a cubic relation, y = a*x^3 + b*x^2 + c*x + d, compute 
m12, m23, m34, M13, and M24 as above. Then

   a = (M24-M13)/(x4-x1)
   b = M13 - a*(x1+x2+x3),
   c = m12 - a*(x1^2+x1*x2+x2^2) - b*(x1+x2)
   d = y1 - a*x1^3 - b*x1^2 - c*x1

To find an exponential relation, y = c*a^x, take the quotient of 
two y-values y2/y1 = a^(x2-x1), and take its (x2-x1) root 
(y2/y1)^(1/[x2-x1]). That should be the constant a. Do the same for 
other pairs of lines in the table. Then c = y1/a^x1. If these values 
are not constant, then there is no exponential relation.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Equations, Graphs, Translations

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