Question

use the coordinates of the plotted points to complete the calculation below. pay attention to negative signs.
slope = \frac{change\\ in\\ y}{change\\ in\\ x} = \frac{\square - (-2)}{4 - \square}
= \frac{\square}{\square}

Explanation:

Step1: Identify the two points

The two points on the line are \((2, 2)\) and \((4, 1)\) (from the graph). Wait, no, looking at the graph, maybe the points are \((2, 2)\) and \((4, 1)\)? Wait, the y - coordinate of the first point (let's say point Q) is 2, x - coordinate 2; the other point (let's say point A) has x - coordinate 4, y - coordinate 1? Wait, no, the slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take the two points: let's say \((x_1,y_1)=(2,2)\) and \((x_2,y_2)=(4,1)\)? Wait, no, maybe I misread. Wait, the change in y: if one point is (2,2) and the other is (4,1), then change in y is \(1 - 2=-1\)? Wait, no, the problem has "change in y" as \(\square- 2\), so the first y - coordinate is 2, and the second y - coordinate is \(\square\). Let's look at the x - coordinate: "change in x" is \(4-\square\), so the first x - coordinate is \(\square\), second is 4. So the two points are \((x_1,y_1)\) and \((x_2,y_2)=(4,y_2)\), with \(y_1 = 2\), \(x_1\) is the first x. From the graph, the two points are (2,2) and (4,1)? Wait, no, maybe (2,2) and (4,1) is wrong. Wait, let's re - examine. The slope formula is \(m=\frac{\text{change in }y}{\text{change in }x}=\frac{y_2 - y_1}{x_2 - x_1}\). The problem has \(\text{slope}=\frac{\text{change in }y}{\text{change in }x}=\frac{\square - 2}{4-\square}\). So \(y_2=\square\), \(y_1 = 2\); \(x_2 = 4\), \(x_1=\square\). From the graph, the two points are (2,2) and (4,1). So \(y_2 = 1\), \(x_1 = 2\). So change in y is \(1-2=-1\), change in x is \(4 - 2 = 2\). Wait, but the problem's "change in y" is \(\square-2\), so \(\square=1\) (since \(1 - 2=-1\)), and "change in x" is \(4-\square\), so \(\square = 2\) (since \(4 - 2 = 2\)). Then the slope is \(\frac{1 - 2}{4 - 2}=\frac{-1}{2}\). Wait, maybe I made a mistake. Wait, let's take the two points as (2,2) and (4,1). Then:

Step2: Fill in the change in y

Change in y is \(y_2 - y_1\). If \(y_1 = 2\) and \(y_2=1\), then change in y is \(1 - 2\). So the first box in the change in y numerator is 1.

Step3: Fill in the change in x

Change in x is \(x_2 - x_1\). If \(x_1 = 2\) and \(x_2 = 4\), then change in x is \(4 - 2\). So the box in the change in x denominator is 2.

Step4: Calculate the slope

Slope \(=\frac{1 - 2}{4 - 2}=\frac{-1}{2}\). Wait, but let's check again. The two points are (2,2) and (4,1). So \(y_2 = 1\), \(y_1 = 2\), \(x_2 = 4\), \(x_1 = 2\). So change in y: \(1-2=-1\), change in x: \(4 - 2 = 2\). So the first box (change in y numerator) is 1, the box in change in x denominator is 2, and then the slope is \(\frac{-1}{2}\) (or \(\frac{1 - 2}{4 - 2}=\frac{-1}{2}\)).

Answer:

First, for the "change in y" numerator: \(1\) (since \(1-2\) gives the change in y), for the "change in x" denominator: \(2\) (since \(4 - 2\) gives the change in x), and the slope is \(\frac{-1}{2}\) (or \(\frac{1 - 2}{4 - 2}=\frac{-1}{2}\)). So the filled boxes are: change in y numerator: \(1\), change in x denominator: \(2\), and the slope is \(\frac{-1}{2}\) (or \(\frac{1 - 2}{4 - 2}=\frac{-1}{2}\)).