In Exercises 37–40, find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (-1, -2) and (-3, -4)

Verified step by step guidance

1. Step 1: Recall the formula for the slope of a line passing through two points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁).
2. Step 2: Identify the coordinates of the two points given in the problem: Point 1 is (-1, -2) and Point 2 is (-3, -4). Assign x₁ = -1, y₁ = -2, x₂ = -3, and y₂ = -4.
3. Step 3: Substitute the values into the slope formula: m = ((-4) - (-2)) / ((-3) - (-1)). Simplify the numerator and denominator separately.
4. Step 4: Simplify the numerator: (-4) - (-2) becomes -4 + 2. Simplify the denominator: (-3) - (-1) becomes -3 + 1.
5. Step 5: Determine whether the slope is positive, negative, zero, or undefined. Based on the sign of the slope, decide whether the line rises, falls, is horizontal, or is vertical.

Key Concepts

Slope of a Line

The slope of a line measures its steepness and direction, calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points. It is represented by the formula m = (y2 - y1) / (x2 - x1). A positive slope indicates the line rises, while a negative slope indicates it falls.

Undefined Slope

A slope is considered undefined when the line is vertical, meaning the x-coordinates of the two points are the same. In this case, the formula for slope results in division by zero, which is mathematically undefined. Vertical lines do not rise or fall but run straight up and down.

Line Orientation

The orientation of a line can be categorized as rising, falling, horizontal, or vertical based on its slope. A rising line has a positive slope, a falling line has a negative slope, a horizontal line has a slope of zero, and a vertical line has an undefined slope. Understanding these orientations helps in visualizing the relationship between the points.