Which graph represents a function with a rate of change of 0.5?

Question

On a coordinate plane, a line with negative slope goes through points (negative 1, 1) and (0, negative 1).
On a coordinate plane, a line with negative slope goes through points (negative 2, 0) and (0, negative 1).
On a coordinate plane, a line with positive slope goes through points (0, negative 1) and (1, 1).
On a coordinate plane, a line with positive slope goes through points (0, negative 1) and (2, 0).

Answer 1

To determine which graph represents a function with a rate of change of 0.5, we need to calculate the slope of each line represented by the points given. The slope (rate of change) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's evaluate each option:

1. First line: points (-1, 1) and (0, -1)

   • \(y_2 = -1, y_1 = 1, x_2 = 0, x_1 = -1\)
   • Slope = \(\frac{-1 - 1}{0 - (-1)} = \frac{-2}{1} = -2\)

2. Second line: points (-2, 0) and (0, -1)

   • \(y_2 = -1, y_1 = 0, x_2 = 0, x_1 = -2\)
   • Slope = \(\frac{-1 - 0}{0 - (-2)} = \frac{-1}{2} = -0.5\)

3. Third line: points (0, -1) and (1, 1)

   • \(y_2 = 1, y_1 = -1, x_2 = 1, x_1 = 0\)
   • Slope = \(\frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2\)

4. Fourth line: points (0, -1) and (2, 0)

   • \(y_2 = 0, y_1 = -1, x_2 = 2, x_1 = 0\)
   • Slope = \(\frac{0 - (-1)}{2 - 0} = \frac{1}{2} = 0.5\)

The only line that has a rate of change of 0.5 is the fourth line, which goes through the points (0, -1) and (2, 0).