Asked by Daniel Gibson
Change the slope or y-intercept value in the equation below to fix the Marble slide so we can clear the star
Answers
Answered by
GPT-4o mini
To adjust the slope or y-intercept of a linear equation to ensure a marble slide clears a star, you need to understand the impact of these values:
1. **Slope (m)**: This determines how steep the line is. Increasing the slope means the line rises more quickly, while decreasing it makes the slide less steep.
2. **Y-intercept (b)**: This is the point where the line crosses the y-axis. Adjusting the y-intercept moves the line up or down without changing its slope.
### Example Scenario
If you have an equation of the line given by:
\[ y = mx + b \]
To clear a star situated at a point (x, y):
1. **If the marble is falling short of the star** (the line is below the star):
- **Increase** the y-intercept \( b \) to lift the entire slide up.
- **Increase** the slope \( m \) to make the slide steeper, which might also help hit the target higher.
2. **If the marble is overshooting the star** (the line is above the star):
- **Decrease** the y-intercept \( b \) to lower the entire slide.
- **Decrease** the slope \( m \) to make the slide less steep, which will help in hitting lower targets.
### Steps to Adjust
1. Identify the current values of \( m \) and \( b \).
2. Determine the current trajectory and position of the star.
3. Decide on the necessary adjustments based on the above criteria.
4. Rewrite the equation with the new slope and/or y-intercept.
If you provide the specific equation or details about the marble slide and the star's position, I can give you a more precise adjustment!
1. **Slope (m)**: This determines how steep the line is. Increasing the slope means the line rises more quickly, while decreasing it makes the slide less steep.
2. **Y-intercept (b)**: This is the point where the line crosses the y-axis. Adjusting the y-intercept moves the line up or down without changing its slope.
### Example Scenario
If you have an equation of the line given by:
\[ y = mx + b \]
To clear a star situated at a point (x, y):
1. **If the marble is falling short of the star** (the line is below the star):
- **Increase** the y-intercept \( b \) to lift the entire slide up.
- **Increase** the slope \( m \) to make the slide steeper, which might also help hit the target higher.
2. **If the marble is overshooting the star** (the line is above the star):
- **Decrease** the y-intercept \( b \) to lower the entire slide.
- **Decrease** the slope \( m \) to make the slide less steep, which will help in hitting lower targets.
### Steps to Adjust
1. Identify the current values of \( m \) and \( b \).
2. Determine the current trajectory and position of the star.
3. Decide on the necessary adjustments based on the above criteria.
4. Rewrite the equation with the new slope and/or y-intercept.
If you provide the specific equation or details about the marble slide and the star's position, I can give you a more precise adjustment!