1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Mathematical Modeling Direct, inverse, joint variations; Least squares regression

1. y varies directly as x. 2. y is directly proportional to x. 3. y = mx for some nonzero constant m. NOTE: m is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find m. y = mx yields 3 = m(2) or m = 1.5. Thus, y = 1.5x. Direct Variation Statements

1. y varies directly as the nth power of x. 2. y is directly proportional to the nth power of x. 3. y = kx n for some nonzero constant k. NOTE: k is the constant of variation or constant of proportionality. Direct Variation as nth Power

1. y varies inversely as x. 2. y is inversely proportional to x. 3. y = k / x for some nonzero constant k. NOTE: k is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find k. y = k / x yields 3 = k / 2 or k = 6. Thus, y = 6 / x. Inverse Variation Statements

1. z varies jointly as x and y. 2. z is jointly proportional to x and y. 3. y = kxy for some nonzero constant k. NOTE: k is the constant of variation. Example: If z = 15 when x = 2 and y = 3, find k. y = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5. Thus, y = 2.5xy. Joint Variation Statements

This method is used to find the “best fit” straight line y = ax + b for a set of points, (x,y), in the x-y coordinate plane. Least Squares Regression

The “best fit” straight line, y = ax + b, for a set of points, (x,y), in the x-y coordinate plane. Least Squares Regression Line

XYX 2 XY  Least Squares Regression Line Solving for a = 0.57 and b = 3, yields y = 0.57x + 3.