Question

analyzing student work
charlie used a regression calculator to generate the equation ( f(x) = -0.15x + 20.1 ) for the ordered pairs ( (2, 15) ), ( (4, 21) ), ( (6, 26) ), ( (8, 20) ), and ( (10, 14) ).
is a linear representation the best way to represent the data? if it is, explain why. if not, explain why and suggest a better alternative.

Explanation:

Step1: Analyze the data trend

Plot the ordered pairs \((2,15)\), \((4,21)\), \((6,26)\), \((8,20)\), \((10,14)\). Observe the changes in \(y\)-values.

Step2: Check linearity

For a linear function \(y = mx + b\) (here \(f(x)=- 0.15x + 20.1\), \(m=-0.15\)), the \(y\)-values should change at a constant rate. But the data first increases (\(15\) to \(21\) to \(26\)) and then decreases (\(26\) to \(20\) to \(14\)). A linear model assumes a constant rate of change (either always increasing or always decreasing).

Step3: Suggest an alternative

A quadratic model \(y = ax^{2}+bx + c\) (where \(a
eq0\)) can capture a maximum - like behavior (the data first increases then decreases, which is characteristic of a parabola opening downwards if \(a<0\)).

Answer:

A linear representation is not the best way to represent the data. The data first increases and then decreases. A linear model assumes a constant rate of change (either always increasing or always decreasing). A quadratic model would be a better alternative as it can account for the non - linear (increasing then decreasing) trend in the data.