a (graph of a line through the origin), b (table with x: 13,16,21,27 and y:39,48,63,81), c (graph of a curve in the first quadrant), d (table with x:3,5,18,21 and y:6,10,36,42)

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Accepted Answer

Assuming the question is to identify which of A, B, C, D represents a proportional relationship (direct variation, $ y = kx $ where $ k $ is constant). Let's analyze each:

### Option A (Graph):
The graph is a straight line through the origin, but it has two segments? Wait, no—wait, the line passes through (0,0), but the left segment goes from (-2, -4) to (0,0)? Wait, slope: from (0,0) to (1, 4)? Wait, no, maybe I misread. Wait, if it's a straight line through the origin, but does it have a constant slope? Wait, no—wait, the left part: from x=-2, y=-4? Wait, no, the graph in A: let's check points. If x=1, y=4? Wait, maybe the slope is 4? But wait, the left segment: when x=-1, y=-4? No, maybe the graph is two lines? Wait, no, maybe it's a single line. Wait, no—actually, a proportional relationship must be a straight line through the origin with constant slope. But wait, the graph in A: does it pass through (0,0) and have constant slope? Wait, maybe the left part is a different line? Wait, no, maybe the graph is not a function? Wait, no, maybe I made a mistake. Let's check others.

### Option B (Table):
Calculate $ \frac{y}{x} $:
- $ \frac{39}{13} = 3 $
- $ \frac{48}{16} = 3 $
- $ \frac{63}{21} = 3 $
- $ \frac{81}{27} = 3 $
All ratios are 3 (constant). So $ y = 3x $, proportional.

### Option C (Graph):
The graph is a curve (hyperbola-like), so it's not a straight line. Not proportional (proportional is linear, straight line through origin).

### Option D (Table):
Calculate $ \frac{y}{x} $:
- $ \frac{6}{3} = 2 $
- $ \frac{10}{5} = 2 $
- $ \frac{36}{18} = 2 $
- $ \frac{42}{21} = 2 $
Wait, all ratios are 2? Wait, but the graph in A: wait, maybe I misanalyzed A. Wait, D's table: $ y = 2x $, so proportional. But wait, the graph in A: is it a straight line through origin? Wait, maybe the graph in A is not a function (fails vertical line test)? Because at x=0, it has two points? Wait, the graph in A: the left segment goes down to (-2, -4) and the right goes up to (2, 4)? Wait, no, the vertical line at x=0 would intersect twice (once from the left, once from the right), so it's not a function. So A is not a function, hence not a proportional relationship (which requires a function $ y = kx $).

Wait, but D's table: $ y = 2x $, proportional. B's table: $ y = 3x $, proportional. Wait, but maybe the graph in A is a function? Wait, no—if the graph has two lines crossing at the origin, it's not a function (fails vertical line test). So A is invalid (not a function). C is a curve (not linear). Now, B and D: both have constant $ \frac{y}{x} $. Wait, but maybe the question is which is NOT proportional? Wait, no, maybe I misread. Wait, let's recheck:

Wait, the graph in A: is it a single line? Wait, maybe the left part is a different line, but actually, the graph in A: if you look at the axes, the left segment is from (-2, -4) to (0,0), and the right from (0,0) to (2, 4). So it's two lines, but together they form a straight line? Wait, no—from (-2, -4) to (0,0): slope is $ \frac{0 - (-4)}{0 - (-2)} = 2 $. From (0,0) to (2, 4): slope is $ \frac{4 - 0}{2 - 0} = 2 $. Wait, so slope is 2? Then why does the right part look steeper? Maybe the graph is scaled. Wait, but if the slope is 2, then $ y = 2x $. But the table in D is $ y = 2x $, and B is $ y = 3x $. Wait, but the graph in A: if it's a straight line with slope 2, passing through origin, then it is proportional. But the table in D is also proportional. Wait, maybe the graph in A is not a function (two lines, so vertical line test fails). Because at x=0, it's a single point? Wait, no—if the two segments meet at (0,0), then it's a single line (a straight line with slope 2, from (-2, -4) to (2, 4), passing through (0,0)). So it is a function? Wait, no—for a function, each x has one y. If x=0, y=0 (only one point). The left segment: x from -2 to 0, y from -4 to 0. The right segment: x from 0 to 2, y from 0 to 4. So it's a function (each x has one y). Then slope is 2 (from (-2, -4) to (0,0): slope 2; (0,0) to (2,4): slope 2). So A is $ y = 2x $, proportional. B is $ y = 3x $, proportional. D is $ y = 2x $, proportional. C is not. Wait, this is confusing. Maybe the original question is to find which is NOT proportional. Let's re-express:

Proportional relationship: $ y = kx $, so graph is straight line through origin, and $ \frac{y}{x} = k $ (constant) for all x.

- **A**: Graph is straight line through origin, slope constant (2), so proportional.
- **B**: $ \frac{y}{x} = 3 $ (constant), proportional.
- **C**: Graph is a curve (hyperbola), not linear, so NOT proportional.
- **D**: $ \frac{y}{x} = 2 $ (constant), proportional.

So if the question is "Which does NOT represent a proportional relationship?", the answer is C.

But since the user didn't specify the question, assuming it's to identify non-proportional, the answer is C.

# Brief Explanations:
- **A**: Straight line through origin (proportional).
- **B**: $ \frac{y}{x} = 3 $ (constant, proportional).
- **C**: Curve (not linear, non - proportional).
- **D**: $ \frac{y}{x} = 2 $ (constant, proportional).

# Answer:
C

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QUESTION IMAGE

Question

Explanation:

Option A (Graph):

Option B (Table):

Option C (Graph):

Option D (Table):

Answer: