1. **Problem Statement:** Given two points on a line with coordinates $(x_1, y_1)$ and $(x_2, y_2)$, we need to find: a) The vertical change between the points. b) The horizontal change between the points. c) The rate of change of $y$ with respect to $x$ (also called the slope). 2. **Step a: Vertical Change** The vertical change is the difference in the $y$-coordinates of the two points. It is computed as: $$\text{Vertical change} = y_2 - y_1$$ This tells us how much the $y$ value changes when moving from the first point to the second. 3. **Step b: Horizontal Change** The horizontal change is the difference in the $x$-coordinates of the two points. It is computed as: $$\text{Horizontal change} = x_2 - x_1$$ This tells us how much the $x$ value changes when moving from the first point to the second. 4. **Step c: Rate of Change (Slope)** The rate of change of $y$ with respect to $x$ is the ratio of the vertical change to the horizontal change. This is also called the slope of the line connecting the two points: $$\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$$ This value tells us how steep the line is and the direction it goes (upwards if positive, downwards if negative). **Final answers:** a) Vertical change = $y_2 - y_1$ b) Horizontal change = $x_2 - x_1$ c) Rate of change (slope) = $\frac{y_2 - y_1}{x_2 - x_1}$