Question

recognizing graphs of proportional relationships
circle all the problems with graphs that do not represent a proportional relationship. for the problems that are circled, explain why the graphs do not represent a proportional relationship.

1. graph 1
2. graph 2
3. graph 3
4. graph 4
5. graph 5
6. graph 6

Explanation:

To determine if a graph represents a proportional relationship, we use the following rules:

1. A proportional relationship has the form \( y = kx \), where \( k \) is the constant of proportionality.
2. The graph of a proportional relationship is a straight line that passes through the origin \((0,0)\).

Problem 1:

Step 1: Check if the graph passes through the origin.

The graph in Problem 1 intersects the \( y \)-axis at \( (0, 3) \), not \( (0, 0) \).

Step 2: Conclusion.

Since it does not pass through the origin, it is not a proportional relationship.

Problem 2:

Step 1: Check if the graph passes through the origin.

The graph in Problem 2 passes through \( (0, 0) \) and is a straight line.

Step 2: Conclusion.

It represents a proportional relationship (no circle).

Problem 3:

Step 1: Check if the graph passes through the origin.

The graph in Problem 3 passes through \( (0, 0) \) and is a straight line.

Step 2: Conclusion.

It represents a proportional relationship (no circle).

Problem 4:

Step 1: Check if the graph passes through the origin.

The graph in Problem 4 intersects the \( x \)-axis at \( (1, 0) \), not \( (0, 0) \).

Step 2: Conclusion.

Since it does not pass through the origin, it is not a proportional relationship.

Problem 5:

Step 1: Check if the graph is a straight line.

The graph in Problem 5 is a curve (not a straight line), even though it passes through \( (0, 0) \).

Step 2: Conclusion.

A proportional relationship must be linear (straight line). Thus, it is not a proportional relationship.

Problem 6:

Step 1: Check if the graph is a straight line through the origin.

The graph in Problem 6 is a straight line passing through \( (0, 0) \).

Step 2: Conclusion.

It represents a proportional relationship (no circle).

Answer:

• Circle Problem 1: The graph does not pass through the origin \((0,0)\) (intercepts \( y \)-axis at \( (0, 3) \)).
• Circle Problem 4: The graph does not pass through the origin \((0,0)\) (intercepts \( x \)-axis at \( (1, 0) \)).
• Circle Problem 5: The graph is not a straight line (it is a curve), so it cannot represent \( y = kx \).

Problems 2, 3, and 6 do not need to be circled (they represent proportional relationships).