Question
describe the vertical translation between y =5x+2 and y=-7x 2 points below the equation y=-7x y=5x+2 -7 points below the equation y=-7x 2 pints above the equation y=-7x
Answers
GPT-4o mini
To describe the vertical translation between the lines represented by the equations \(y = 5x + 2\) and \(y = -7x\), we need to analyze their properties and the nature of their intersections.
1. **Equations and Slopes**:
- The line represented by \(y = 5x + 2\) has a slope of 5 and a y-intercept of 2. This means it rises quickly as \(x\) increases.
- The line represented by \(y = -7x\) has a slope of -7 and a y-intercept of 0. This means it falls quickly as \(x\) increases.
2. **Visualizing Vertical Translation**:
- Vertical translation refers to moving a line up or down without changing its slope. This can be understood in terms of how far one line can be moved to align with or relate to the other.
3. **Finding Specific Points**:
- **Points Below \(y = -7x\)**:
- We can find the points on the line \(y = 5x + 2\) that are two points below the line \(y = -7x\).
- For example, if \(x = 0\):
- On \(y = -7x\): \(y = -7(0) = 0\).
- Two points below this would be \(-2\), so at \(x = 0\), we have the point \((0, -2)\).
- For another value of \(x = 1\):
- On \(y = -7(1) = -7\).
- Two points below would be \(-9\) resulting in the point \((1, -9)\).
- **Points Above \(y = -7x\)**:
- Similarly, if we take points above the line \(y = -7x\) (for \(x = 0\):
- The point on \(y = -7x\) is \(0\).
- Two points above this would be \(2\); hence at \(x = 0\), we have \((0, 2)\).
- And for \(x = 1\):
- The point is \(-7\).
- Two points above this would be \(-5\) resulting in the point \((1, -5)\).
4. **Determining Vertical Distance**:
- The vertical distance should be calculated by finding the difference in the y-values of the two lines at specific points (where they can be compared). The vertical distance where the two lines intersect can also illustrate how far one line is displaced from the other.
In summary, the vertical translation between the two lines can be characterized by observing points that fall above and below the line \(y = -7x\) with selected x-values.
1. **Equations and Slopes**:
- The line represented by \(y = 5x + 2\) has a slope of 5 and a y-intercept of 2. This means it rises quickly as \(x\) increases.
- The line represented by \(y = -7x\) has a slope of -7 and a y-intercept of 0. This means it falls quickly as \(x\) increases.
2. **Visualizing Vertical Translation**:
- Vertical translation refers to moving a line up or down without changing its slope. This can be understood in terms of how far one line can be moved to align with or relate to the other.
3. **Finding Specific Points**:
- **Points Below \(y = -7x\)**:
- We can find the points on the line \(y = 5x + 2\) that are two points below the line \(y = -7x\).
- For example, if \(x = 0\):
- On \(y = -7x\): \(y = -7(0) = 0\).
- Two points below this would be \(-2\), so at \(x = 0\), we have the point \((0, -2)\).
- For another value of \(x = 1\):
- On \(y = -7(1) = -7\).
- Two points below would be \(-9\) resulting in the point \((1, -9)\).
- **Points Above \(y = -7x\)**:
- Similarly, if we take points above the line \(y = -7x\) (for \(x = 0\):
- The point on \(y = -7x\) is \(0\).
- Two points above this would be \(2\); hence at \(x = 0\), we have \((0, 2)\).
- And for \(x = 1\):
- The point is \(-7\).
- Two points above this would be \(-5\) resulting in the point \((1, -5)\).
4. **Determining Vertical Distance**:
- The vertical distance should be calculated by finding the difference in the y-values of the two lines at specific points (where they can be compared). The vertical distance where the two lines intersect can also illustrate how far one line is displaced from the other.
In summary, the vertical translation between the two lines can be characterized by observing points that fall above and below the line \(y = -7x\) with selected x-values.