Question

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 negative because more points are below the line than above the line.
she should expect the slope to be positive because the line has a positive y - intercept.
she should expect the slope to be positive because all the plotted points have positive coordinates.

Explanation:

Step1: Recall the slope - direction relationship

The slope of a line \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If a line tilts down from left - to - right (\(x_2>x_1\) and \(y_2 < y_1\)), then \(m=\frac{y_2 - y_1}{x_2 - x_1}<0\).

Step2: Analyze each option

• Option 1:

A line that tilts down from left - to - right has a negative slope. Mathematically, if we take two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line with \(x_2>x_1\) (moving from left to right) and \(y_2 < y_1\), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Since \(y_2 - y_1<0\) and \(x_2 - x_1>0\), \(m<0\).

• Option 2:

The number of points above or below the line is related to the goodness of fit of the trend - line (how well the line represents the data points), not to the sign of the slope.

• Option 3:

The \(y\) - intercept (\(b\) in the equation \(y = mx + b\)) and the slope (\(m\)) are two independent parameters. A line can have a positive \(y\) - intercept (\(b>0\)) and a negative slope (\(m<0\)) (e.g., \(y=-2x + 5\)).

• Option 4:

The fact that all the plotted points have positive coordinates (\(x>0,y>0\)) has no relation to the sign of the slope. For example, the line \(y = 2x+1\) (positive slope) and \(y=-2x + 3\) (negative slope) can both pass through points with positive \(x\) and \(y\) values.

Answer:

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