Question

8.3.3 quiz: nonlinear models
which regression equation best fits these data?

a. $y=-0.32x^{2}-1.26x + 15.81$
b. $y = 0.32x^{2}+1.26x + 15.81$
c. $y=-0.67x + 12.41$

Explanation:

Step1: Analyze the shape of the data

The data points do not form a straight - line (so a linear regression \(y = mx + b\) like option C (\(y=-0.67x + 12.41\)) is less likely). Since the data has a non - linear (parabolic - like) shape (the \(y\) values first increase and then decrease as \(x\) moves from \(-6\) to \(4\)), we consider quadratic regression equations (\(y=ax^{2}+bx + c\)).

Step2: Check the coefficient of \(x^{2}\)

For a quadratic function \(y = ax^{2}+bx + c\), if \(a>0\), the parabola opens upwards (\(y\) has a minimum value), and if \(a < 0\), the parabola opens downwards (\(y\) has a maximum value). Looking at the data: when \(x=-3\) and \(x = - 1\), \(y = 17\); as \(x\) moves from \(-3\) to \(4\), \(y\) decreases. So the parabola opens downwards, which means \(a<0\).

Step3: Evaluate the options

Option B has \(a = 0.32>0\) (parabola opens upwards), so we can eliminate option B. Option A has \(a=-0.32<0\) (parabola opens downwards).

Answer:

A. \(y=-0.32x^{2}-1.26x + 15.81\)