Question

8.3.3 quiz: nonlinear models
what is the quadratic regression equation that fits these data?
a. $y = 12.09cdot1.01^{x}$
b. $y = 2.09x^{2}+0.33x + 3.06$
c. $y = 2.84x^{2}+0.68x + 1.12$

Explanation:

Step1: Recall quadratic regression form

A quadratic regression equation has the form \(y = ax^{2}+bx + c\). Option A is an exponential function (\(y = ab^{x}\) form), so we can eliminate it.

Step2: Test \(x = 0\)

For a quadratic equation \(y=ax^{2}+bx + c\), when \(x = 0\), \(y=c\).
Looking at the data, when \(x = 0\), \(y = 2\).
For option B: when \(x = 0\), \(y=2.09\times0^{2}+0.33\times0 + 3.06=3.06\)
For option C: when \(x = 0\), \(y=2.84\times0^{2}+0.68\times0+1.12 = 1.12\) (This step shows that we check the \(y -\)intercept property of the quadratic function. Since the data has \(y = 2\) at \(x = 0\), neither B nor C is a perfect fit from this simple check. But if we assume that there is a calculation - based quadratic regression (using a calculator or software for quadratic regression on the data set \((x,y)\) pairs: \((-4,35),(-3,20),(-2,12),(-1,6),(0,2),(1,6),(2,10),(3,24),(4,38)\)):
We input the \(x\) and \(y\) values into a quadratic regression calculator (many graphing calculators or online tools like Excel's Data Analysis Toolpak can perform quadratic regression).
The general formula for quadratic regression: Given \(n\) data points \((x_{i},y_{i})\) for \(i = 1,\cdots,n\), we want to minimize \(S=\sum_{i = 1}^{n}(y_{i}-(ax_{i}^{2}+bx_{i}+c))^{2}\)
By performing the quadratic regression on the data set (using a calculator or software), we find the coefficients \(a\approx2.09\), \(b\approx0.33\), \(c\approx3.06\)

Answer:

B. \(y = 2.09x^{2}+0.33x + 3.06\)