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Understanding how to plot points on a coordinate plane is essential for graphing lines and analyzing their properties. A coordinate plane is divided into four quadrants by a horizontal line called the x-axis and a vertical line called the y-axis. The point where these axes intersect is known as the origin, designated as (0,0).

When plotting the points given in the exercise, such as (0,0) and (8,-4), you start at the origin. For (0,0), you are already at the origin, so no movement is necessary. For (8,-4), you move 8 units to the right along the x-axis, which represents the horizontal coordinate, and then 4 units down along the y-axis, representing the vertical coordinate. This process visually represents the location of the points, serving as a foundation for finding the slope of the line that passes through them.

The slope of a line is a numerical measure of its steepness, often represented by the letter 'm'. It can be positive, negative, zero, or undefined. The formula to find the slope between two points is given as
\( m = \frac{y2 - y1}{x2 - x1} \).

Here’s what each part of the formula represents:

• \( m \) is the slope of the line.
• \( y2 - y1 \) measures the vertical change between two points on the y-axis.
• \( x2 - x1 \) measures the horizontal change between the points on the x-axis.

If the line moves up as it goes right, the slope is positive. If it moves down as it goes right, the slope is negative. A horizontal line has a slope of zero, and a vertical line has an undefined slope. The slope formula is a vital tool for describing the direction and steepness of a line.

To calculate the slope using the given points (0,0) and (8,-4), we apply the slope formula. We designate (0,0) as Point A (with coordinates \( x1 = 0 \) and \( y1 = 0 \)), and (8,-4) as Point B (with coordinates \( x2 = 8 \) and \( y2 = -4 \)).

Substituting these values into the slope formula, we get:
\[ m = \frac{y2 - y1}{x2 - x1} = \frac{-4 - 0}{8 - 0} = -\frac{4}{8} = -\frac{1}{2} \].
The slope is calculated to be -1/2, indicating that for every 2 units the line moves horizontally to the right, it moves down 1 unit. It confirms that our line is inclined downward from left to right, which matches the negative sign of the slope. Through this process, students can not only find the slope but also understand the direction in which the line is moving on the graph.