A function is considered proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant. This means that the graph of the function is a straight line passing through the origin (0,0).
Let's analyze the given functions:
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\( y = x + 12 \) - This is not proportional because it does not pass through the origin (when \( x = 0 \), \( y = 12 \)).
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\( y = x + \frac{1}{2} \) - This is also not proportional because it does not pass through the origin (when \( x = 0 \), \( y = \frac{1}{2} \)).
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\( y = x^2 \) - This is not proportional because it is a quadratic function and does not have the form \( y = kx \).
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\( y = \frac{1}{2}x \) - This is proportional because it can be expressed in the form \( y = kx \) with \( k = \frac{1}{2} \). It passes through the origin.
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\( y = 12x \) - This is also proportional because it can be expressed in the form \( y = kx \) with \( k = 12 \). It passes through the origin.
Thus, the proportional functions from the list are:
- \( y = \frac{1}{2}x \)
- \( y = 12x \)
Both of these functions are proportional.