Question

1. does the graph show a proportional relationship? why or why not?

Explanation:

To determine if a graph shows a proportional relationship, we use the definition of a proportional relationship: a relationship between two quantities where the ratio of the two quantities (also known as the constant of proportionality, \( k \)) is constant, and the graph is a straight line that passes through the origin \((0,0)\).

Step 1: Recall the properties of a proportional relationship

A proportional relationship between two variables \( x \) (independent) and \( y \) (dependent) can be represented by the equation \( y = kx \), where \( k \) is a constant (the constant of proportionality). For a graph to represent a proportional relationship:

1. The graph must be a straight line (linear).
2. The line must pass through the origin \((0,0)\).

Step 2: Analyze the graph (hypothetical example)

Since the actual graph is not visible, we use the general rules:

• If the graph is a straight line and passes through \((0,0)\), then it is proportional.
• If the graph is not a straight line, or is a straight line that does not pass through \((0,0)\), then it is not proportional.

Example Application (if the graph were, e.g., a straight line through the origin):

Suppose the graph is a straight line passing through \((0,0)\) (e.g., \( y = 2x \)). Then:

• The ratio \( \frac{y}{x} = 2 \) (constant), so it is proportional.

Example Application (if the graph were, e.g., a straight line not through the origin):

Suppose the graph is a straight line like \( y = 2x + 3 \) (which does not pass through \((0,0)\), since when \( x=0 \), \( y=3 \)). Then:

• The ratio \( \frac{y}{x} = \frac{2x + 3}{x} = 2 + \frac{3}{x} \), which is not constant (it changes with \( x \)). Thus, it is not proportional.

Final Answer (General Form):

To determine if the graph shows a proportional relationship, check two things:

1. Is the graph a straight line?
2. Does the line pass through the origin \((0,0)\)?

• If both are true: Yes, it is proportional (because the ratio \( \frac{y}{x} \) is constant, and the equation is \( y = kx \)).
• If either is false: No, it is not proportional (because the ratio \( \frac{y}{x} \) is not constant, or the line does not satisfy \( y = kx \)).

(Note: For a specific graph, substitute the actual shape and intercepts into this reasoning.)

Answer:

To determine if a graph shows a proportional relationship, we use the definition of a proportional relationship: a relationship between two quantities where the ratio of the two quantities (also known as the constant of proportionality, \( k \)) is constant, and the graph is a straight line that passes through the origin \((0,0)\).

Step 1: Recall the properties of a proportional relationship

A proportional relationship between two variables \( x \) (independent) and \( y \) (dependent) can be represented by the equation \( y = kx \), where \( k \) is a constant (the constant of proportionality). For a graph to represent a proportional relationship:

1. The graph must be a straight line (linear).
2. The line must pass through the origin \((0,0)\).

Step 2: Analyze the graph (hypothetical example)

Since the actual graph is not visible, we use the general rules:

• If the graph is a straight line and passes through \((0,0)\), then it is proportional.
• If the graph is not a straight line, or is a straight line that does not pass through \((0,0)\), then it is not proportional.

Example Application (if the graph were, e.g., a straight line through the origin):

Suppose the graph is a straight line passing through \((0,0)\) (e.g., \( y = 2x \)). Then:

• The ratio \( \frac{y}{x} = 2 \) (constant), so it is proportional.

Example Application (if the graph were, e.g., a straight line not through the origin):

Suppose the graph is a straight line like \( y = 2x + 3 \) (which does not pass through \((0,0)\), since when \( x=0 \), \( y=3 \)). Then:

• The ratio \( \frac{y}{x} = \frac{2x + 3}{x} = 2 + \frac{3}{x} \), which is not constant (it changes with \( x \)). Thus, it is not proportional.

Final Answer (General Form):

To determine if the graph shows a proportional relationship, check two things:

1. Is the graph a straight line?
2. Does the line pass through the origin \((0,0)\)?

• If both are true: Yes, it is proportional (because the ratio \( \frac{y}{x} \) is constant, and the equation is \( y = kx \)).
• If either is false: No, it is not proportional (because the ratio \( \frac{y}{x} \) is not constant, or the line does not satisfy \( y = kx \)).

(Note: For a specific graph, substitute the actual shape and intercepts into this reasoning.)