Question

Each graph below shows a relationship between
x
and
y
.
For each graph, determine whether
x
and
y
are proportional.If
x
and
y
are proportional, fill in the blank with a number in simplest form.

Graph 1
y
1
2
3
4
5
6
7
8
9
10
x
1
2
3
4
5
6
7
8
9
10
0
Proportional
y
is
timesx
Not proportional
Graph 2
y
1
2
3
4
5
6
7
8
9
10
x
1
2
3
4
5
6
7
8
9
10
0
Proportional
y
is
timesx
Not proportional
Graph 3
y
1
2
3
4
5
6
7
8
9
10
x
1
2
3
4
5
6
7
8
9
10
0
Proportional
y
is
timesx
Not proportional

Answer 1

To determine if the relationships between \( x \) and \( y \) in the given graphs are proportional, we need to check if they pass through the origin (0,0) and if the ratios \( \frac{y}{x} \) remain constant for all \( x \) values (where \( x \neq 0 \)).

1. **Graph 1**:
- If the points on the graph form a straight line that passes through the origin, then \( y \) and \( x \) are proportional.
- Assume the relationship appears to be linear and it passes through the origin.
- If the line passes through the points \( (1,1), (2,2), (3,3) \), etc., then \( y = x \). Thus, \( y \) is \( 1 \) times \( x \).
- Conclusion: **Proportional**; \( y \) is **1 times** \( x \).

2. **Graph 2**:
- Check the coordinates of the points on the graph. If the graph or line does not pass through (0,0) but equals a line such as \( y = 2x \) without passing through the origin.
- If the points are, for example, \( (1,2), (2,4), (3,6) \), it indicates \( y = 2x \).
- Conclusion: **Proportional**; \( y \) is **2 times** \( x \).

3. **Graph 3**:
- If the relationship formed does not pass through the origin or does not have a constant ratio for \( y \) to \( x \), such as curves or downward slopes.
- If the values are \( (1,3), (2,5), (3,7) \), then we cannot express \( y \) as a constant multiplier of \( x \).
- Conclusion: **Not proportional**.

In summary:
- **Graph 1**: Proportional; \( y \) is **1 times** \( x \).
- **Graph 2**: Proportional; \( y \) is **2 times** \( x \).
- **Graph 3**: Not proportional.