Question

select three true statements about this graph. a it is proportional because when the x values go up by 1, the y values go up by 1.99. b it is not proportional because the graph does not include (0,0). c the point (1, 1.99) shows the price per carton. d the point (3, 5.97) shows that the price for 5.97 cartons is $3. e the point (3, 5.97) shows that the price of 3 cartons is $5.97.

Explanation:

Step1: Analyze Option A

A proportional relationship has a constant ratio (slope) and passes through \((0,0)\). If \(x\) increases by 1 and \(y\) by 1.99, the ratio \(\frac{y}{x}\) is constant (\(1.99\) for \(x = 1\), \(3.98\) for \(x = 2\)? Wait, no—wait, if it's proportional, \(y=kx\), so when \(x = 1\), \(y=k\); \(x = 2\), \(y = 2k\), etc. So if \(x\) increases by 1, \(y\) increases by \(k\). So if the rate is constant (like a linear graph with slope \(k\) and passing through \((0,0)\)), it's proportional. Wait, but let's check the other options first.

Step2: Analyze Option B

A proportional relationship's graph must pass through \((0,0)\) (since \(y=kx\) implies when \(x = 0\), \(y = 0\)). If the graph does not include \((0,0)\), it's not proportional. So B could be true, but wait—maybe the graph is linear but not proportional? Wait, no—proportional is a special case of linear (with \(y\)-intercept 0). So if it's not passing through \((0,0)\), it's not proportional. But let's check other options.

Step3: Analyze Option C

The point \((1, 1.99)\): if \(x\) is the number of cartons and \(y\) is the price, then when \(x = 1\) (1 carton), \(y = 1.99\) (price), so that's the price per carton. So C is true.

Step4: Analyze Option D

The point \((3, 5.97)\): \(x\) is the number of cartons, \(y\) is the price. So \(x = 3\) cartons, \(y = 5.97\) dollars. D says "price for 5.97 cartons is $3"—that's reversed. So D is false.

Step5: Analyze Option E

The point \((3, 5.97)\): \(x = 3\) (cartons), \(y = 5.97\) (price). So it shows 3 cartons cost $5.97. E is true.

Wait, but earlier about A: if the graph is linear with slope 1.99 and passes through \((0,0)\), then it's proportional. Wait, maybe the graph is a straight line through \((0,0)\)? Wait, the problem is about a graph (probably a line) with points like \((1,1.99)\), \((2,3.98)\), \((3,5.97)\), etc. Let's check the ratio: \(1.99/1 = 1.99\), \(3.98/2 = 1.99\), \(5.97/3 = 1.99\). So the ratio \(y/x\) is constant (1.99), so it is proportional. Wait, then why does B say it's not proportional because it doesn't include \((0,0)\)? Maybe the graph does include \((0,0)\)? Wait, maybe the original graph (not shown) has \((0,0)\)? Wait, the options are conflicting. Wait, let's re-express:

If the graph is a straight line passing through \((0,0)\) and has a constant ratio \(y/x = 1.99\), then it is proportional. So A says "it is proportional because when the x values go up by 1, the y values go up by 1.99"—that's correct because the slope is 1.99, and since it's proportional (passes through \((0,0)\)), the rate of change is constant. Then B says it's not proportional because it doesn't include \((0,0)\)—but if the ratio is constant and it does include \((0,0)\), then B is false. Wait, maybe the graph is about price per carton, so when \(x = 0\) (0 cartons), price is $0, so \((0,0)\) is included. Then B is false.

So let's re-evaluate:

• A: If the graph is linear with slope 1.99 and passes through \((0,0)\) (so proportional), then when \(x\) increases by 1, \(y\) increases by 1.99 (since slope is 1.99). So A is true.

• C: \((1, 1.99)\) means 1 carton costs $1.99, so that's the price per carton. True.

• E: \((3, 5.97)\) means 3 cartons cost $5.97. True.

Let's check the ratios: \(1.99/1 = 1.99\), \(5.97/3 = 1.99\), so the ratio is constant, so it is proportional (so A is true, B is false). D is false (reversed x and y). So the three true statements are A, C, E.

Answer:

A. It is proportional because when the x values go up by 1, the y values go up by 1.99.
C. The point (1, 1.99) shows the price per carton.
E. The point (3, 5.97) shows that the price of 3 cartons is $5.97.