Question

which function is the best fit for the data?
$y=-2x + 9$
$y=-4x + 12$
$y=2x + 8$
$y=-x + 5$

Explanation:

Step1: Analyze the slope of the data trend

The data points show a decreasing trend (as \(x\) increases, \(y\) decreases), so the slope \(m\) of the best - fit line should be negative. We can eliminate \(y = 2x+8\) since its slope \(m = 2>0\).

Step2: Estimate the slope magnitude

Let's take two approximate points. Suppose we consider a point near \((1,7)\) and a point near \((3,3)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((x_1,y_1)=(1,7)\) and \((x_2,y_2)=(3,3)\), we have \(m=\frac{3 - 7}{3 - 1}=\frac{-4}{2}=- 2\).

Step3: Check the \(y\) - intercept

For \(y=-2x + 9\), when \(x = 0\), \(y=9\). For \(y=-4x + 12\), when \(x = 0\), \(y = 12\). For \(y=-x + 5\), when \(x = 0\), \(y=5\). If we consider the general position of the data points, the \(y\) - value when \(x = 0\) (extrapolating the trend) is closer to \(y = 9\) than to \(y = 12\) or \(y = 5\).

Answer:

\(y=-2x + 9\)