When graphing a linear equation that is written in the form `y = mx+b,` the variable m represents the slope of the line and b represents the y-intercept. Assuming that `b gt 0,` which of the following best describes how reversing the sign of b will affect the graph ?

Options

A. The new line will be shifted down b units.
B. The new line will be shifted down 2b units.
C. the new line will be perfect reflection across the x-axis.
D. The new line will be a perfect reflection across the y-axis.

Explanation

To solve the problem, we need to analyze how reversing the sign of the y-intercept \( b \) in the linear equation \( y = mx + b \) affects the graph of the line.

### Step-by-Step Solution:

1. **Understand the Original Equation**:
The original equation is given as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.

2. **Identify the Y-Intercept**:
Since \( b > 0 \), the line crosses the y-axis above the origin (0,0). This means the point (0, b) is above the x-axis.

3. **Reversing the Sign of \( b \)**:
When we reverse the sign of \( b \), we replace it with \( -b \). The new equation becomes \( y = mx - b \).

4. **Determine the New Y-Intercept**:
In the new equation \( y = mx - b \), the y-intercept is now at the point (0, -b). Since \( b \) is positive, \( -b \) will be negative, indicating that the new line crosses the y-axis below the origin.

5. **Visualize the Shift**:
The original line was at (0, b), and now it has moved to (0, -b). This represents a downward shift of \( 2b \) units from the original position because it moves from \( b \) (above the x-axis) to \( -b \) (below the x-axis).

6. **Conclusion**:
Therefore, the graph of the line shifts down by \( 2b \) units when the sign of \( b \) is reversed.

### Final Answer:
The new line will be shifted down \( 2b \) units.