Certainly! Let's graph the linear equation \( y = -x \).

### Understanding the Equation
The equation \( y = -x \) is a straight line with the following characteristics:

- **Slope (m):** The coefficient of \( x \) is -1. This means the line decreases by 1 unit on the \( y \)-axis for every 1 unit increase on the \( x \)-axis.
- **Y-intercept (b):** The point where the line crosses the \( y \)-axis is at (0, 0).

### Steps to Graph \( y = -x \)

1. **Plot the Y-intercept:**
   - Start by plotting the point (0, 0) on the graph.

2. **Use the Slope to Find Another Point:**
   - The slope is -1, which can be interpreted as "down 1, right 1."
   - From (0, 0), move right 1 unit to (1, 0) and then down 1 unit to (1, -1). Plot the point (1, -1).

3. **Draw the Line:**
   - Connect the two points with a straight line.
   - Extend the line in both directions, adding arrows to indicate it continues indefinitely.

4. **Additional Points (Optional):**
   - For accuracy, you can plot more points using the slope.
   - From (0, 0), move left 1 unit to (-1, 0) and then up 1 unit to (-1, 1). Plot the point (-1, 1).

### Visual Representation

Here's a simple ASCII representation of the graph:

```
    y
     |
  2 -|        *
     |
  1 -|      *
     |
  0 -|----*----*---- x
     |  *
 -1 -|*
     |
 -2 -|
     |
```

- **Points Plotted:**
  - (0, 0)
  - (1, -1)
  - (-1, 1)

### Description of the Graph

- The line passes through the origin (0,0).
- It has a negative slope, meaning it descends from left to right.
- For every step you take to the right on the \( x \)-axis, you move one step down on the \( y \)-axis.
- The line extends infinitely in both directions.

### Slope-Intercept Form

The equation \( y = -x \) is already in the **slope-intercept form**, which is:

\[
y = mx + b
\]

Where:
- \( m \) is the slope.
- \( b \) is the y-intercept.

For \( y = -x \):
- \( m = -1 \)
- \( b = 0 \)

### Conclusion

Graphing \( y = -x \) results in a straight line that passes through the origin with a slope of -1. By plotting key points and connecting them, you can visualize this linear relationship effectively.

If you have access to graphing tools or software (like Desmos, GeoGebra, or even graph paper), you can plot this line more accurately by following the steps outlined above.

Home Question Bank Math Geometry

Question

Graph the line \( y=-x \)

Ask by Reid Romero. in the United States

Jan 30,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

To graph the line \( y = -x \), plot the origin (0,0) and another point like (1, -1). Draw a straight line through these points, extending in both directions.

Solution

Certainly! Let's graph the linear equation \( y = -x \). ### Understanding the Equation The equation \( y = -x \) is a straight line with the following characteristics: - **Slope (m):** The coefficient of \( x \) is -1. This means the line decreases by 1 unit on the \( y \)-axis for every 1 unit increase on the \( x \)-axis. - **Y-intercept (b):** The point where the line crosses the \( y \)-axis is at (0, 0). ### Steps to Graph \( y = -x \) 1. **Plot the Y-intercept:** - Start by plotting the point (0, 0) on the graph. 2. **Use the Slope to Find Another Point:** - The slope is -1, which can be interpreted as "down 1, right 1." - From (0, 0), move right 1 unit to (1, 0) and then down 1 unit to (1, -1). Plot the point (1, -1). 3. **Draw the Line:** - Connect the two points with a straight line. - Extend the line in both directions, adding arrows to indicate it continues indefinitely. 4. **Additional Points (Optional):** - For accuracy, you can plot more points using the slope. - From (0, 0), move left 1 unit to (-1, 0) and then up 1 unit to (-1, 1). Plot the point (-1, 1). ### Visual Representation Here's a simple ASCII representation of the graph: ``` y | 2 -| * | 1 -| * | 0 -|----*----*---- x | * -1 -|* | -2 -| | ``` - **Points Plotted:** - (0, 0) - (1, -1) - (-1, 1) ### Description of the Graph - The line passes through the origin (0,0). - It has a negative slope, meaning it descends from left to right. - For every step you take to the right on the \( x \)-axis, you move one step down on the \( y \)-axis. - The line extends infinitely in both directions. ### Slope-Intercept Form The equation \( y = -x \) is already in the **slope-intercept form**, which is: \[ y = mx + b \] Where: - \( m \) is the slope. - \( b \) is the y-intercept. For \( y = -x \): - \( m = -1 \) - \( b = 0 \) ### Conclusion Graphing \( y = -x \) results in a straight line that passes through the origin with a slope of -1. By plotting key points and connecting them, you can visualize this linear relationship effectively. If you have access to graphing tools or software (like Desmos, GeoGebra, or even graph paper), you can plot this line more accurately by following the steps outlined above.

Reviewed and approved by the UpStudy tutoring team

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Extra Insights

To graph the line \( y = -x \), start by identifying key points. When \( x = 0 \), \( y = 0 \) (the origin point). When \( x = 1 \), \( y = -1 \) gives the point (1, -1). Also, for \( x = -1 \), \( y = 1 \) results in (-1, 1). Plot these points on a coordinate plane and then draw a straight line through them, extending it in both directions. You’ll see that the line slopes downward from left to right, illustrating a perfect diagonal split of the first and third quadrants. To make it even more engaging, consider that this equation represents a relationship where every increase in \( x \) leads to a decrease in \( y \), showcasing the beauty of linear relationships in mathematics! Plus, the line \( y = -x \) is a reflection over the origin, making it a perfect example of symmetry in the Cartesian plane.

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