Graph the line with the equation $$ y=-\frac{1}{2} x-1 $$

Question

UpStudy AI · Accepted Answer

Sure! Let's graph the line given by the equation:

$$ y = -\frac{1}{2}x - 1 $$

### Understanding the Equation

This equation is in **slope-intercept form**, which is:

$$ y = mx + b $$

- **$$ m $$** is the **slope** of the line.
- **$$ b $$** is the **y-intercept**, the point where the line crosses the y-axis.

For your equation:

- **Slope ($$ m $$)**: $$ -\frac{1}{2} $$
- **Y-intercept ($$ b $$)**: $$ -1 $$

### Plotting the Line

1. **Plot the Y-intercept ($$ 0, -1 $$)**

Start by placing a point at $$ (0, -1) $$ on the graph.

2. **Use the Slope to Find Another Point**

The slope $$ -\frac{1}{2} $$ means that for every 2 units you move **right** (positive direction on the x-axis), you move **down** by 1 unit (negative direction on the y-axis).

- Starting from $$ (0, -1) $$:
     - Move right 2 units to $$ x = 2 $$.
     - Move down 1 unit to $$ y = -2 $$.
   - This gives you the point $$ (2, -2) $$.

3. **Draw the Line**

Connect the two points with a straight line extending in both directions.

### Table of Values

Here's a simple table to determine multiple points on the line:

| $$ x $$ | $$ y = -\frac{1}{2}x - 1 $$ |
|-------|-----------------------------|
| -2    | $$ -\frac{1}{2}(-2) - 1 = 1 - 1 = 0 $$ → $$ (-2, 0) $$ |
| 0     | $$ -1 $$ → $$ (0, -1) $$ |
| 2     | $$ -2 $$ → $$ (2, -2) $$ |
| 4     | $$ -3 $$ → $$ (4, -3) $$ |

### ASCII Representation of the Graph

While it's limited, here's a simple ASCII graph to visualize the line:

```
 y
 ^
 |
3|          
2|          *
1|      *    
0|*---------*--------->
               2    4    x
```

- The points plotted are:
  - $$ (-2, 0) $$
  - $$ (0, -1) $$
  - $$ (2, -2) $$
  - $$ (4, -3) $$

### Visual Plot

For a more precise and visual plot, you can use graphing tools like [Desmos](https://www.desmos.com/calculator) or any graphing calculator:

1. Go to [Desmos Graphing Calculator](https://www.desmos.com/calculator).
2. Enter the equation $$ y = -\frac{1}{2}x - 1 $$ into the input field.
3. Observe the graph, which will display the line passing through the points you've plotted.

### Summary

- **Slope ($$ m $$)**: The line decreases by $$ \frac{1}{2} $$ unit vertically for every 1 unit it moves horizontally.
- **Y-intercept ($$ b $$)**: The line crosses the y-axis at $$ (0, -1) $$.
- **Points to Plot**: $$ (-2, 0) $$, $$ (0, -1) $$, $$ (2, -2) $$, $$ (4, -3) $$.

By plotting these points and drawing a straight line through them, you'll have an accurate graph of the equation $$ y = -\frac{1}{2}x - 1 $$.
To graph the line $$ y = -\frac{1}{2}x - 1 $$, plot the y-intercept at $$ (0, -1) $$. Then, use the slope $$ -\frac{1}{2} $$ to find another point by moving 2 units right and 1 unit down from the y-intercept, arriving at $$ (2, -2) $$. Draw a straight line through these points to complete the graph.

Graph the line with the equation

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