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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

1 answer

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.