Linear Function Through Points (5½,10) and (½,5): Coordinate Analysis

To determine the nature of the linear function, let's calculate the slope of the line passing through the given points:

• Given points: $(x_1, y_1) = \left(\frac{1}{2}, 5\right)$ and $(x_2, y_2) = \left(5\frac{1}{2}, 10\right)$.
• Use the formula for the slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 5}{5.5 - 0.5} = \frac{5}{5} = 1.$
• The slope $m = 1$ is positive, indicating the line is increasing from left to right.

Hence, the function is a bottom-up function, indicating it increases as the x-values increase.

Therefore, the correct answer is: Bottom-up function.