Casio fx-CG10 Advanced Operation Manual page 46

Fostering Advanced Algebraic Thinking with Casio Technology
Sample Solution—Douglas Fir Trees (continued)
2
Using the r values, the quadratic function was chosen to model the data.
However, as the circumference of a tree increases, it would seem the height of
the tree should also increase. This means that either we should limit the
domain of the function or that the data would be modeled best by a function
that is strictly increasing—either the linear or power regression function, and
not the quadratic.
The quadratic regression function models the data well, but only for trees that
have a circumference between approximately 0 and 9 meters.
Discussion to share with students: A regression function is often used to
predict values both within the range of the data set (interpolation) and
beyond the range of the data set (extrapolation). The quadratic regression is
particularly useful for interpolation, but not extrapolation. The quadratic
regression does not allow for extrapolation, predicting values outside of the
range of the data set, as it predicts the height of a tree 12 meters in
circumference to be shorter than a tree that is 8 meters in circumference. For
this reason, using the quadratic regression function to predict the height of
the trees with circumference of 2 and 9 meters is thought to be much more
accurate than predicting the height of trees with a circumference greater than
9 meters. In order to be able to both interpolate and extrapolate, we will
choose the power regression function (instead of the quadratic) to be a better
model of the data.
So, let's copy the power regression function, and use this equation instead to
approximate the height of Douglas fir trees with a circumference of 2, 8, and
12 meters.