Graph the equation. $$ y=\frac{1}{2} x-6 $$

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UpStudy AI · Accepted Answer

Certainly! Let's graph the linear equation:

$$ y = \frac{1}{2}x - 6 $$

### Understanding the Equation

The given equation is in **slope-intercept form**, which is:

$$ y = mx + b $$

Where:
- $$ m $$ is the **slope** of the line.
- $$ b $$ is the **y-intercept** (the point where the line crosses the y-axis).

For the equation $$ y = \frac{1}{2}x - 6 $$:
- **Slope ($$ m $$)**: $$ \frac{1}{2} $$
- **Y-intercept ($$ b $$)**: $$ -6 $$

### Step-by-Step Graphing

1. **Plot the Y-Intercept ($$ b $$)**

The y-intercept is the point where the line crosses the y-axis (where $$ x = 0 $$).

$$
   \text{Point: } (0, -6)
   $$

2. **Use the Slope ($$ m $$) 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 **up** by 1 unit (positive direction on the y-axis).

Starting from $$ (0, -6) $$:
   - Move **right** 2 units to $$ x = 2 $$.
   - Move **up** 1 unit to $$ y = -5 $$.

$$
   \text{Second Point: } (2, -5)
   $$

3. **Plot a Third Point for Accuracy**

Alternatively, you can choose $$ x = 4 $$ to find another point:

$$
   y = \frac{1}{2}(4) - 6 = 2 - 6 = -4
   $$
   
   $$
   \text{Third Point: } (4, -4)
   $$

4. **Draw the Line**

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

### Table of Values

| $$ x $$ | $$ y = \frac{1}{2}x - 6 $$ |
|:------:|:--------------------------:|
|  -4    |            -8              |
|  -2    |            -7              |
|   0    |            -6              |
|   2    |            -5              |
|   4    |            -4              |
|   6    |            -3              |

### Visual Representation

Here's an approximate ASCII representation of the graph:

```
y
|
-2 |                     *
-4 |               *
-6 |* 
-8 |
------------------------------ x
    0   2   4   6   8
```

- The **\*** marks represent the points plotted on the graph:
  - $$ (0, -6) $$
  - $$ (2, -5) $$
  - $$ (4, -4) $$

### Key Features of the Graph

- **Slope ($$ \frac{1}{2} $$)**: The line rises by 1 unit for every 2 units it moves to the right.
- **Y-Intercept (-6)**: The line crosses the y-axis at (0, -6).
- **X-Intercept**: To find where the line crosses the x-axis (where $$ y = 0 $$):
  
  $$
  0 = \frac{1}{2}x - 6 \
  \frac{1}{2}x = 6 \
  x = 12
  $$
  
  So, the x-intercept is at $$ (12, 0) $$.

### Final Notes

- This is a **straight line** graph since the equation is linear.
- The line extends infinitely in both directions unless restricted by a specific domain.

If you have graphing software or tools like Desmos, GeoGebra, or a graphing calculator, you can plot these points to see a precise graph.
To graph the equation $$ y = \frac{1}{2}x - 6 $$, plot the y-intercept at (0, -6). Then, use the slope of $$ \frac{1}{2} $$ to find another point, such as (2, -5). Draw a straight line through these points to complete the graph.

Graph the equation. \( y=\frac{1}{2} x-6 \)

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