It is sometimes desired to fit a second degree polynomialE(y|x) = α + βx+ γx2to a set of data so that its stationary point will fall in a given region. Methods are shown to calculatea, b, c, the least squares estimates of the regression coefficients α, β, and γ. An example is given in which this theory is applied to fit a parabolic regression to a set of data so that in a given contiguous and finite region of the independent variable, the regression function is strictly increasing and concave upwards.