Question
Which graph represents a line with a slope of and a y-intercept equal to that of the line y = x – 2?
A coordinate plane with a line starting at (negative 2, negative 4), passing through (0, negative 2) and (2, 1).
A coordinate plane with a line passing through (negative 3, 0) and (0, 2).
A coordinate plane with a line passing through (0, 2) and (2, negative 1).
A coordinate plane with a line passing through (negative 3, 0) and (0, negative 2).
A coordinate plane with a line starting at (negative 2, negative 4), passing through (0, negative 2) and (2, 1).
A coordinate plane with a line passing through (negative 3, 0) and (0, 2).
A coordinate plane with a line passing through (0, 2) and (2, negative 1).
A coordinate plane with a line passing through (negative 3, 0) and (0, negative 2).
Answers
GPT-4o mini
To answer this question, we first need to identify the slope and y-intercept of the line given by the equation \(y = x - 2\).
1. **Identify the slope and y-intercept**:
From the equation \(y = x - 2\), we can see that:
- The slope (m) is 1 (the coefficient of x).
- The y-intercept (b) is -2 (the constant term).
Now that we have identified the slope to be 1 (m=1) and the y-intercept to be -2 (b=-2), we will look for a graph of a line with these characteristics:
2. **Characteristics of the desired line**:
The line will have the following characteristics:
- It will pass through the point (0, -2) because that is the y-intercept.
- It will have a slope of 1, meaning for every unit increase in x, y increases by 1.
3. **Analyzing each option**:
- **Option 1**: A coordinate plane with a line starting at (−2, −4), passing through (0, −2) and (2, 1).
- This line passes through (0, -2), which is correct for the y-intercept. The slope can be calculated by checking the rise/run between points, and it is indeed 1. This option works.
- **Option 2**: A coordinate plane with a line passing through (−3, 0) and (0, 2).
- The y-intercept is at (0, 2), which does not match the required y-intercept of -2. This option does not work.
- **Option 3**: A coordinate plane with a line passing through (0, 2) and (2, -1).
- The y-intercept is at (0, 2), which again does not match our requirement of -2. This option does not work.
- **Option 4**: A coordinate plane with a line passing through (−3, 0) and (0, −2).
- This line passes through (0, -2), matching the required y-intercept. The slope can also be computed, and it is not 1.
Only the **first option** has both the correct y-intercept and slope characteristics.
Thus, the correct graph is the first option: **A coordinate plane with a line starting at (−2, −4), passing through (0, −2) and (2, 1)**.
1. **Identify the slope and y-intercept**:
From the equation \(y = x - 2\), we can see that:
- The slope (m) is 1 (the coefficient of x).
- The y-intercept (b) is -2 (the constant term).
Now that we have identified the slope to be 1 (m=1) and the y-intercept to be -2 (b=-2), we will look for a graph of a line with these characteristics:
2. **Characteristics of the desired line**:
The line will have the following characteristics:
- It will pass through the point (0, -2) because that is the y-intercept.
- It will have a slope of 1, meaning for every unit increase in x, y increases by 1.
3. **Analyzing each option**:
- **Option 1**: A coordinate plane with a line starting at (−2, −4), passing through (0, −2) and (2, 1).
- This line passes through (0, -2), which is correct for the y-intercept. The slope can be calculated by checking the rise/run between points, and it is indeed 1. This option works.
- **Option 2**: A coordinate plane with a line passing through (−3, 0) and (0, 2).
- The y-intercept is at (0, 2), which does not match the required y-intercept of -2. This option does not work.
- **Option 3**: A coordinate plane with a line passing through (0, 2) and (2, -1).
- The y-intercept is at (0, 2), which again does not match our requirement of -2. This option does not work.
- **Option 4**: A coordinate plane with a line passing through (−3, 0) and (0, −2).
- This line passes through (0, -2), matching the required y-intercept. The slope can also be computed, and it is not 1.
Only the **first option** has both the correct y-intercept and slope characteristics.
Thus, the correct graph is the first option: **A coordinate plane with a line starting at (−2, −4), passing through (0, −2) and (2, 1)**.