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To determine the residuals for each point on this scatter plot with a line of fit, let's go through each step carefully. Residuals are calculated by finding the difference between the actual \( y \)-value of each point and the \( y \)-value predicted by the line of fit at the corresponding \( x \)-value. Let’s proceed with each point:

1. **Identify the Equation of the Line of Fit:**  
   The line appears to pass through points \((0,2)\) and \((5,6)\). To find the slope \( m \) of the line:
   \[
   m = \frac{6 - 2}{5 - 0} = \frac{4}{5} = 0.8
   \]
   So, the equation of the line is approximately:
   \[
   y = 0.8x + 2
   \]

2. **Calculate Residuals for Each Point**:  
   For each point on the scatter plot, substitute the \( x \)-value into the equation of the line to find the predicted \( y \)-value, then subtract this predicted \( y \)-value from the actual \( y \)-value to get the residual.

Let's go through each data point:

- **Point (0,2)**:  
  Predicted \( y = 0.8(0) + 2 = 2 \)  
  Actual \( y = 2 \)  
  Residual = \( 2 - 2 = 0 \)

- **Point (1,3)**:  
  Predicted \( y = 0.8(1) + 2 = 2.8 \)  
  Actual \( y = 3 \)  
  Residual = \( 3 - 2.8 = 0.2 \)

- **Point (2,2)**:  
  Predicted \( y = 0.8(2) + 2 = 3.6 \)  
  Actual \( y = 2 \)  
  Residual = \( 2 - 3.6 = -1.6 \)

- **Point (3,5)**:  
  Predicted \( y = 0.8(3) + 2 = 4.4 \)  
  Actual \( y = 5 \)  
  Residual = \( 5 - 4.4 = 0.6 \)

- **Point (4,4)**:  
  Predicted \( y = 0.8(4) + 2 = 5.2 \)  
  Actual \( y = 4 \)  
  Residual = \( 4 - 5.2 = -1.2 \)

- **Point (5,6)**:  
  Predicted \( y = 0.8(5) + 2 = 6 \)  
  Actual \( y = 6 \)  
  Residual = \( 6 - 6 = 0 \)

3. **Enter Residual Values**:  
   Based on the calculations above, here are the residuals:

   - (0) \(0\)
   - (1) \(0.2\)
   - (2) \(-1.6\)
   - (3) \(0.6\)
   - (4) \(-1.2\)
   - (5) \(0\)

Would you like further details on any step or explanation on residual interpretation?

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Here are five additional questions for further exploration:

1. What does a positive residual indicate about a data point in relation to the line of fit?
2. How can you tell if a line of fit is a good model based on residuals?
3. What would happen to residuals if the line of fit were adjusted?
4. How can residuals be used to improve predictive models?
5. Can you calculate residuals for nonlinear models?

**Tip:** Residuals can help identify patterns in data that a simple linear model might miss, especially if there is a systematic trend in the residuals.

Use the scatterplot and line of fit to calculate residuals for the points shown and enter them.

Learn how to calculate residuals for each point on a scatterplot with a line of fit. This example includes finding the line equation, predicting y-values, and calculating the difference between actual and predicted values for each point. Suitable for students in Grades 8-10.

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