Question
A two-dimensional Cartesian coordinate system is shown. The x-axis and y-axis are labelled accordingly. A light olive-green line extends from the bottom left quadrant to the upper right quadrant. The line passes through the origin (0,0) and intersects the y-axis at the point (0, 3). The gridlines are evenly spaced, and the axes are clearly marked with tick marks. No other elements are present on the graph.
Which equation matches the function shown in the graph?
Responses
A y = x + 4y = x + 4
B y = 4x + 2y = 4x + 2
C y = 2x + 4y = 2x + 4
D y = -2x + 4
Which equation matches the function shown in the graph?
Responses
A y = x + 4y = x + 4
B y = 4x + 2y = 4x + 2
C y = 2x + 4y = 2x + 4
D y = -2x + 4
Answers
GPT-4o mini
To find the equation of the line described, we can use the point-slope form of the linear equation. The line passes through the origin (0,0) and intersects the y-axis at the point (0,3). This means the line has a y-intercept of 3.
Since it also passes through (0, 0), we can determine the slope (m) using the two points:
1. The origin (0, 0)
2. The point (0, 3)
However, here we only need to find the slope, which can be derived from the intercepts given.
If the line intersects the y-axis at (0, 3), it means that as x is increased, y decreases indicating a negative slope.
To check each equation:
A. \( y = x + 4 \) - This would intersect the y-axis at \( y = 4 \), not \( y = 3 \).
B. \( y = 4x + 2 \) - This would intersect the y-axis at \( y = 2 \), not \( y = 3 \).
C. \( y = 2x + 4 \) - This would also intersect the y-axis at \( y = 4 \), not \( y = 3 \).
D. \( y = -2x + 4 \) - This intersects the y-axis at \( y = 4 \) and decreases as x increases.
As we see that none of the options intersect the y-axis at \( (0, 3) \), it’s evident there's an inconsistency in the provided responses concerning the condition of the intercept at (0, 3).
However, analyzing the form of these equations is essential. Given that the description states that the light olive-green line intersects both the origin and has the point (0,3), the right conclusion should compute a factor representing 3 when the x is set to zero.
Reflecting on the right form ought to derive a slope that yields x when changing direction vertically to hit the intercept.
Based on this review:
The line described should definitely follow a behavior modeled as above. However, since nothing matches to the intercept of 3 directly, the line should have been \( y = mx + 3 \).
If we're constrained to the responses given as A, B, C, and D. None directly satisfy the factual statement best given its slope behavior. Hence, upon selection and checks, these could represent relative forms isolates as options. Although with further analysis into your own coordinates could yield modification.
But with noted matches against slope details, with some adjustment to an overall view, none seem to reflect effectively based on defined positional tabs and stated conditions. A verification of potential queries is necessary for better simplification to check set values.
The best possible relation to a negative slope identifying point drop would then be closely aligned to option D; given conditions!
Since it also passes through (0, 0), we can determine the slope (m) using the two points:
1. The origin (0, 0)
2. The point (0, 3)
However, here we only need to find the slope, which can be derived from the intercepts given.
If the line intersects the y-axis at (0, 3), it means that as x is increased, y decreases indicating a negative slope.
To check each equation:
A. \( y = x + 4 \) - This would intersect the y-axis at \( y = 4 \), not \( y = 3 \).
B. \( y = 4x + 2 \) - This would intersect the y-axis at \( y = 2 \), not \( y = 3 \).
C. \( y = 2x + 4 \) - This would also intersect the y-axis at \( y = 4 \), not \( y = 3 \).
D. \( y = -2x + 4 \) - This intersects the y-axis at \( y = 4 \) and decreases as x increases.
As we see that none of the options intersect the y-axis at \( (0, 3) \), it’s evident there's an inconsistency in the provided responses concerning the condition of the intercept at (0, 3).
However, analyzing the form of these equations is essential. Given that the description states that the light olive-green line intersects both the origin and has the point (0,3), the right conclusion should compute a factor representing 3 when the x is set to zero.
Reflecting on the right form ought to derive a slope that yields x when changing direction vertically to hit the intercept.
Based on this review:
The line described should definitely follow a behavior modeled as above. However, since nothing matches to the intercept of 3 directly, the line should have been \( y = mx + 3 \).
If we're constrained to the responses given as A, B, C, and D. None directly satisfy the factual statement best given its slope behavior. Hence, upon selection and checks, these could represent relative forms isolates as options. Although with further analysis into your own coordinates could yield modification.
But with noted matches against slope details, with some adjustment to an overall view, none seem to reflect effectively based on defined positional tabs and stated conditions. A verification of potential queries is necessary for better simplification to check set values.
The best possible relation to a negative slope identifying point drop would then be closely aligned to option D; given conditions!