Which graph shows a proportional relationship that passes through the point (2, 1)?

Question

The graph of a line that begins at the origin and passes through the point 1 comma 2. The graph of a line that begins at the origin and passes through the point 2 comma 1. The graph of a line that begins at the point zero comma 2 and passes through the point 1 comma 3. The graph of a line that begins at the point zero comma 1 and passes through the point 1 comma 2.

Answer 1

A proportional relationship is characterized by a straight line that passes through the origin (0, 0). In addition, for a relationship to be proportional and to pass through a specific point (x, y), the coordinates must satisfy the linear equation $y=kx$ for some constant $k$.

Given the point (2, 1), we can determine the slope $k$ as follows:

1. Using the coordinates of the point (2, 1), we can find the constant of proportionality $k$: $$k=\frac{y}{x}=\frac{1}{2}$$

This means that the equation of the proportional relationship that passes through (2, 1) is: $$y=\frac{1}{2}x$$

Now, looking at the options:

1. The graph of a line that begins at the origin and passes through (1, 2): This line would not pass through (2, 1), and does not represent a proportional relationship.
2. The graph of a line that begins at the origin and passes through (2, 1): This is the correct choice, as it represents a proportional relationship and passes through the point (2, 1).
3. The graph of a line that begins at (0, 2) and passes through (1, 3): This line does not pass through the origin and therefore is not a proportional relationship.
4. The graph of a line that begins at (0, 1) and passes through (1, 2): This line also does not pass through the origin and thus does not represent a proportional relationship.

So, the correct answer is: The graph of a line that begins at the origin and passes through the point (2, 1).