Question
Which graph displays the linear equation with a y
-intercept of −3
and an ordered pair of (−1,−2)
?(1 point)
Responses
1.A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrows passes through left parenthesis 0 comma negative 3 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrows passes through left parenthesis 0 comma negative 3 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
2.A Cartesian coordinate plane has both the axes ranging from negative 4 to 4 in increments of 1. A decreasing line passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma negative 3 right parenthesis.
Image with alt text: A Cartesian coordinate plane has both the axes ranging from negative 4 to 4 in increments of 1. A decreasing line passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma negative 3 right parenthesis.
3.A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
4.A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis negative 1 comma 1 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis negative 1 comma 1 right parenthesis.
-intercept of −3
and an ordered pair of (−1,−2)
?(1 point)
Responses
1.A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrows passes through left parenthesis 0 comma negative 3 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrows passes through left parenthesis 0 comma negative 3 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
2.A Cartesian coordinate plane has both the axes ranging from negative 4 to 4 in increments of 1. A decreasing line passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma negative 3 right parenthesis.
Image with alt text: A Cartesian coordinate plane has both the axes ranging from negative 4 to 4 in increments of 1. A decreasing line passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma negative 3 right parenthesis.
3.A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis.
4.A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis negative 1 comma 1 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 4 to 4 in unit increments. A line with bi-directional arrowheads passes through left parenthesis negative 3 comma 0 right parenthesis and left parenthesis negative 1 comma 1 right parenthesis.
Answers
Answered by
GPT-4o mini
To determine which graph represents the linear equation with a y-intercept of -3 and passes through the ordered pair (-1, -2), we can derive the equation of the line.
1. The y-intercept is -3, so the equation will be in the form \( y = mx - 3 \), where \( m \) is the slope.
2. We can find the slope \( m \) using the point (-1, -2).
- Substituting this point into the equation:
\[
-2 = m(-1) - 3
\]
\[
-2 + 3 = -m
\]
\[
1 = -m
\]
\[
m = -1
\]
So the equation of our line is:
\[
y = -x - 3
\]
Next, we check the points we calculated against the graph options given:
- **Graph 1**: The line passes through (0, -3) (the correct y-intercept) and (1, -1). The slope between these two points can be calculated as:
\[
\text{slope} = \frac{-1 - (-3)}{1 - 0} = \frac{2}{1} = 2 \quad (\text{not } -1)
\]
So, this graph does not match.
- **Graph 2**: The line passes through (-3, 0) and (0, -3). The slope is:
\[
\text{slope} = \frac{-3 - 0}{0 - (-3)} = \frac{-3}{3} = -1 \quad (\text{correct but does not pass through } (-1, -2))
\]
So, this graph does not match.
- **Graph 3**: The line passes through (-3, 0) and (1, -1). The slope is:
\[
\text{slope} = \frac{-1 - 0}{1 - (-3)} = \frac{-1}{4} \quad (\text{not } -1)
\]
So, this graph does not match.
- **Graph 4**: The line passes through (-3, 0) and (-1, 1). The slope is:
\[
\text{slope} = \frac{1 - 0}{-1 - (-3)} = \frac{1}{2} \quad (\text{not } -1)
\]
So, this graph also does not match.
From our analysis, it seems that none of the given graphs accurately reflect the equation \( y = -x - 3 \) with the specified y-intercept and point.
However, if checking specifically for the correct graph structure matching the intercept and point is essential, considering that only Graph 2 maintains the correct intercept, it appears to be the closest match, despite not passing through (-1, -2).
In this case, you would likely refer to Graph 2 since no option aligns perfectly with both criteria of y-intercept and point but does, at least, indicate the correct intercept alignment. Note that additional visual context might clarify which graph has the intended slope directionality or verifies with explicit points confirmed.
1. The y-intercept is -3, so the equation will be in the form \( y = mx - 3 \), where \( m \) is the slope.
2. We can find the slope \( m \) using the point (-1, -2).
- Substituting this point into the equation:
\[
-2 = m(-1) - 3
\]
\[
-2 + 3 = -m
\]
\[
1 = -m
\]
\[
m = -1
\]
So the equation of our line is:
\[
y = -x - 3
\]
Next, we check the points we calculated against the graph options given:
- **Graph 1**: The line passes through (0, -3) (the correct y-intercept) and (1, -1). The slope between these two points can be calculated as:
\[
\text{slope} = \frac{-1 - (-3)}{1 - 0} = \frac{2}{1} = 2 \quad (\text{not } -1)
\]
So, this graph does not match.
- **Graph 2**: The line passes through (-3, 0) and (0, -3). The slope is:
\[
\text{slope} = \frac{-3 - 0}{0 - (-3)} = \frac{-3}{3} = -1 \quad (\text{correct but does not pass through } (-1, -2))
\]
So, this graph does not match.
- **Graph 3**: The line passes through (-3, 0) and (1, -1). The slope is:
\[
\text{slope} = \frac{-1 - 0}{1 - (-3)} = \frac{-1}{4} \quad (\text{not } -1)
\]
So, this graph does not match.
- **Graph 4**: The line passes through (-3, 0) and (-1, 1). The slope is:
\[
\text{slope} = \frac{1 - 0}{-1 - (-3)} = \frac{1}{2} \quad (\text{not } -1)
\]
So, this graph also does not match.
From our analysis, it seems that none of the given graphs accurately reflect the equation \( y = -x - 3 \) with the specified y-intercept and point.
However, if checking specifically for the correct graph structure matching the intercept and point is essential, considering that only Graph 2 maintains the correct intercept, it appears to be the closest match, despite not passing through (-1, -2).
In this case, you would likely refer to Graph 2 since no option aligns perfectly with both criteria of y-intercept and point but does, at least, indicate the correct intercept alignment. Note that additional visual context might clarify which graph has the intended slope directionality or verifies with explicit points confirmed.