Problem 5 - Algebra 1: Concepts and Skills

Question

Plot the points and draw the line that passes through them. Without finding the slope, determine whether the slope is positive, negative, zero, or undefined. $$ (-2,-2) \text { and }(0,1) $$

Step-by-Step Solution

Answer

After plotting the points and drawing the line, we can see that it rises from left to right as we 'read' the graph in the conventional way. This means the slope of the line is positive.

Step 1: Plot the Points

Start by plotting the points on a graph. Plot the point (-2,-2) and the point (0,1).

Step 2: Draw the Line

Next, draw a line that passes through both of these points.

Step 3: Analyze the Direction of the Line

If the line: \n- Rises from left to right: the slope is positive.\n- Falls from left to right: the slope is negative. \n- Is horizontal: the slope is zero (it doesn't rise or fall).\n- Is vertical: the slope is undefined (the line doesn't move left or right, so we can't calculate rise over run).

Key Concepts

Plotting Points in Coordinate Plane

Understanding how to plot points on a coordinate plane is an essential skill when analyzing the properties of lines. The coordinate plane is a two-dimensional surface divided into four quadrants by a horizontal line (x-axis) and a vertical line (y-axis). Each point is represented by an ordered pair \( (x, y) \) that corresponds to its position along the x-axis and y-axis.

For example, to plot the point \( (-2, -2) \) from the exercise, locate -2 on the x-axis and -2 on the y-axis. The point where these two values meet is where \( (-2, -2) \) is plotted. Similarly, plotting \( (0, 1) \) means you locate 0 on the x-axis, which is the line itself, and go up to 1 on the y-axis. Marking these two points and drawing a line through them sets the foundation for understanding the slope of the line.

Positive and Negative Slope

The slope of a line in a coordinate plane indicates its steepness and direction. When the slope is positive, this means that as you move from left to right along the line, the line rises. Conversely, a negative slope means that the line falls as you move from left to right.

An easy way to remember this is that a positive slope is like climbing uphill, while a negative slope is like descending. The steepness is determined by the value of the slope, which is calculated by the 'rise over run' formula. In a graph, if the line moves upward from left to right, like a line passing through \( (-2, -2) \) and \( (0, 1) \) as in the provided exercise, it signifies a positive slope. If it had moved downward, it would indicate a negative slope.

Undefined and Zero Slope

Lines with zero or undefined slopes are the two special cases in the slope universe. A zero slope is straightforward—it means the line is horizontal. There's no rise or fall as you move along the line; it remains at the same level. This makes the slope equal to zero, hence the name.

In contrast, an undefined slope occurs with a vertical line. This is because slope is normally calculated as a ratio of the vertical change (rise) to the horizontal change (run). However, with vertical lines, there's no horizontal movement (the run is zero), so you're dividing by zero, which is a mathematical no-no. Therefore, the slope is said to be undefined. In our exercise, since the line rises as we move from left to right, the slope is neither zero nor undefined, it is positive.