https://file.aiquickdraw.com/mathbot/file/172669952890224cwu6av.jpg

In the image you provided, the rule for the output (Y) is given by the expression \( Y = -2X - 1 \). Using this expression, we can calculate the output values (Y) based on the input values (X). Let’s fill out the table step-by-step.

### Calculating the Output (Y) for each Input (X):

1. **For X = -1**:
   \[
   Y = -2(-1) - 1 = 2 - 1 = 1
   \]
   So, the output (Y) when \( X = -1 \) is 1.

2. **For X = 0**:
   \[
   Y = -2(0) - 1 = 0 - 1 = -1
   \]
   So, the output (Y) when \( X = 0 \) is -1.

3. **For X = 1**:
   \[
   Y = -2(1) - 1 = -2 - 1 = -3
   \]
   So, the output (Y) when \( X = 1 \) is -3.

4. **For X = 2**:
   \[
   Y = -2(2) - 1 = -4 - 1 = -5
   \]
   So, the output (Y) when \( X = 2 \) is -5.

### Updated Table:
| Input (X) | Expression (Rule) | Output (Y) |
|-----------|-------------------|------------|
| -1        | \( -2x - 1 \)     | 1          |
| 0         | \( -2x - 1 \)     | -1         |
| 1         | \( -2x - 1 \)     | -3         |
| 2         | \( -2x - 1 \)     | -5         |

### Completing the Table of Values:
Now, we fill in the \( (x, y) \) pairs:
\[
(x = -1, y = 1), (x = 0, y = -1), (x = 1, y = -3), (x = 2, y = -5)
\]

You can use these points to plot them on the graph.

Would you like further details on how to plot the points or solve any other part? Here are five follow-up questions you might find helpful:

1. What is the significance of the slope in this equation, \( Y = -2X - 1 \)?
2. How would the graph of this line look in terms of direction (increasing/decreasing)?
3. How do you calculate the slope of a line from two points?
4. What does the y-intercept represent in this equation?
5. How can we extend the table to include more points for a more detailed graph?

**Tip:** When plotting points, remember that a straight line only needs two points to be drawn, but more points can confirm the accuracy of the line.

Complete the input-output table for the rule Y = -2X - 1. Plot the points and complete the graph.

Complete the input-output table for the linear function Y = -2X - 1. Learn how to calculate the outputs based on given inputs and plot the points on a graph. Suitable for students in Grades 6-8.

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