Question

What would the equations y=2/3 x + 1 and y=−2/3 x + 1 look like on a graph?

Answer 1

To visualize the equations \( y = \frac{2}{3}x + 1 \) and \( y = -\frac{2}{3}x + 1 \), we can analyze their characteristics and how they would appear on a graph.

1. **Equation 1: \( y = \frac{2}{3}x + 1 \)**
- **Slope (\( m \))**: The slope is \( \frac{2}{3} \), which means for every 3 units you move to the right (positive x-direction), the line moves up 2 units (positive y-direction).
- **Y-intercept**: The y-intercept is \( 1 \). This is the point where the line crosses the y-axis, at the coordinate (0, 1).
- **Graph Behavior**: This line rises from left to right, starting at (0, 1).

2. **Equation 2: \( y = -\frac{2}{3}x + 1 \)**
- **Slope (\( m \))**: The slope is \( -\frac{2}{3} \), which means for every 3 units you move to the right, the line moves down 2 units.
- **Y-intercept**: The y-intercept is still \( 1 \), so this line also crosses the y-axis at (0, 1).
- **Graph Behavior**: This line falls from left to right, starting at (0, 1).

### Graph Appearance
- Both lines intersect at the point (0, 1).
- The line \( y = \frac{2}{3}x + 1 \) will rise while moving to the right, and the line \( y = -\frac{2}{3}x + 1 \) will fall while moving to the right.
- The two lines are symmetric with respect to the y-axis and create a kind of 'V' shape, opening downwards as they split at the point (0, 1).

### Summary
- The two lines will intersect at (0, 1) and diverge away from each other as you move along the x-axis, forming two linear functions with opposite slopes.