Question

the approximate line of best fit for the given data points, ( y = - 2x - 2 ), is shown on the graph. which is the residual value when ( x = - 2 )? -2 -1 1 2

Explanation:

Step1: Recall residual formula

Residual is actual \( y \) minus predicted \( y \), i.e., \( \text{Residual} = y_{\text{actual}} - y_{\text{predicted}} \).

Step2: Find predicted \( y \) at \( x = -2 \)

Given the line of best fit \( y = -2x - 2 \). Substitute \( x = -2 \):
\( y_{\text{predicted}} = -2(-2) - 2 = 4 - 2 = 2 \).

Step3: Find actual \( y \) at \( x = -2 \)

From the graph, when \( x = -2 \), the actual data point has \( y = 0 \) (since the point is at \( (-2, 0) \)).

Step4: Calculate residual

\( \text{Residual} = y_{\text{actual}} - y_{\text{predicted}} = 0 - 2 = -2 \)? Wait, no—wait, wait, maybe I misread the graph. Wait, looking at the graph: the data point at \( x = -2 \) is on the x-axis, so \( y_{\text{actual}} = 0 \). Wait, but wait, let's check again. Wait, the line at \( x = -2 \): \( y = -2(-2) - 2 = 2 \). The actual point is \( (-2, 0) \)? Wait, no, maybe I made a mistake. Wait, no—wait, the graph: when \( x = -2 \), the data point is at \( y = 0 \)? Wait, no, looking at the grid: the x-axis is horizontal, y-axis vertical. At \( x = -2 \), the data point is on the x-axis, so \( y = 0 \). Then predicted \( y \) is \( -2(-2) - 2 = 2 \). So residual is \( 0 - 2 = -2 \)? But wait, the options include -2, but let me check again. Wait, maybe I misread the actual point. Wait, no—wait, maybe the actual point is at \( y = 0 \), predicted is 2, so residual is \( 0 - 2 = -2 \)? But wait, the options have -2 as an option. Wait, but let me confirm. Wait, the formula for residual is \( \text{Residual} = \text{Observed} - \text{Predicted} \). So observed \( y \) (actual) minus predicted \( y \). So if actual \( y \) is 0, predicted is 2, then residual is \( 0 - 2 = -2 \). But wait, maybe I messed up the actual point. Wait, looking at the graph: the data point at \( x = -2 \) is on the x-axis, so \( y = 0 \). The line at \( x = -2 \) is \( y = 2 \). So residual is \( 0 - 2 = -2 \). So the residual is -2.

Wait, but wait, maybe I made a mistake. Wait, let's re-express:

Residual = Actual \( y \) - Predicted \( y \)

At \( x = -2 \):

• Predicted \( y \): \( y = -2(-2) - 2 = 4 - 2 = 2 \)
• Actual \( y \): From the graph, the data point at \( x = -2 \) is \( (-2, 0) \), so \( y_{\text{actual}} = 0 \)
• Residual: \( 0 - 2 = -2 \)

So the residual is -2.

Answer:

-2