Graph the equation $$ y=-\frac{1}{3} x+2 $$

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UpStudy AI · Accepted Answer

Certainly! Let's graph the linear equation $$ y = -\frac{1}{3}x + 2 $$. This is a straight line in the Cartesian plane. Here's a step-by-step guide to help you plot it:

### 1. Identify the Slope and Y-Intercept

The equation is in the slope-intercept form $$ y = mx + b $$, where:
- **$$ m $$** is the slope of the line.
- **$$ b $$** is the y-intercept (the point where the line crosses the y-axis).

For $$ y = -\frac{1}{3}x + 2 $$:
- **Slope ($$ m $$)**: $$ -\frac{1}{3} $$
- **Y-intercept ($$ b $$)**: $$ 2 $$

### 2. Plot the Y-Intercept

Start by plotting the y-intercept on the graph:
- **Point (0, 2)**: This is where the line crosses the y-axis.

### 3. Use the Slope to Find Another Point

The slope $$ -\frac{1}{3} $$ means that for every 3 units you move horizontally to the right, you move 1 unit down vertically.

Starting from (0, 2):
- Move **3 units to the right** (positive direction on the x-axis).
- Move **1 unit down** (negative direction on the y-axis).

This brings you to the point:
- **Point (3, 1)**

### 4. Draw the Line

Plot both points (0, 2) and (3, 1) on the graph and draw a straight line passing through them. This line represents the equation $$ y = -\frac{1}{3}x + 2 $$.

### 5. Additional Points (Optional)

For greater accuracy, you can plot more points using the same method:
- From (3, 1), move 3 units right to (6, 0)
- From (6, 0), move 3 units right to (9, -1)
- And so on.

### 6. Graph Visualization

While I can't provide an actual graph image here, here's a simple ASCII representation:

```
y
|
3 |         
2 |      * (0,2)
1 |         * (3,1)
0 |------------*---------- x
-1|               * (6,0)
-2|
```

### 7. Summary

- **Slope**: The line decreases because the slope is negative ($$ -\frac{1}{3} $$).
- **Y-Intercept**: The line crosses the y-axis at (0, 2).
- **Direction**: As $$ x $$ increases, $$ y $$ decreases.

### 8. Using Graphing Tools

For a more precise and visual graph, you might consider using online graphing tools such as:
- [Desmos Graphing Calculator](https://www.desmos.com/calculator)
- [GeoGebra](https://www.geogebra.org/graphing)

Simply enter the equation $$ y = -\frac{1}{3}x + 2 $$ into these tools, and they will generate the graph for you.

---

Feel free to ask if you need further assistance with graphing or understanding linear equations!
To graph the equation $$ y = -\frac{1}{3}x + 2 $$, follow these steps: 1. **Plot the Y-Intercept**: Start at (0, 2). 2. **Use the Slope**: Move 3 units right and 1 unit down to reach (3, 1). 3. **Draw the Line**: Connect (0, 2) and (3, 1) with a straight line. This line represents the equation $$ y = -\frac{1}{3}x + 2 $$.

Graph the equation \( y=-\frac{1}{3} x+2 \)

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