Determining Linear RelationsDate: 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/ |
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