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### Determining Linear Relations

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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
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations

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