Question

amani is calculating the slope of the trend line in the scatterplot shown. how can amani use the trend line to check the reasonableness of her answer? she should expect the slope to be negative because the line tilts down moving from left to right on the graph. she should expect the slope to be positive because the line has a positive y - intercept. she should expect the slope to be negative because more points are below the line than above the line. she should expect the slope to be positive because all the plotted points have positive coordinates.

Explanation:

Step1: Recall the slope concept

The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a line moving from left - to - right, if \(y\) decreases as \(x\) increases (\(y_2 x_1\)), the slope is negative.

Step2: Analyze the trend line

In the given scatter - plot, as \(x\) (moving from left to right) increases, the \(y\) - values of the trend line decrease. According to the formula \(m=\frac{\Delta y}{\Delta x}\) (\(\Delta y=y_2 - y_1<0\) and \(\Delta x=x_2 - x_1>0\)), the slope \(m<0\). The sign of the \(y\) - intercept (\(b = 72\) in this case) and the number of points above or below the line do not determine the sign of the slope. The sign of the coordinates of the plotted points also does not determine the sign of the slope. The key is the direction of the line (increasing or decreasing from left - to - right).

Answer:

She should expect the slope to be negative because the line tilts down moving from left to right on the graph.