Problem 65 - Calculus Volume 1

Question

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. $$ (2,4) \text { and }(1,4) $$

Step-by-Step Solution

Answer

The slope is 0; the line is horizontal.

Step 1: Calculate the Change in y (Vertical Change)

Identify the y-coordinates of the given points: the first point is (2,4) and the second point is (1,4). Calculate the difference in y-values: \( y_2 - y_1 = 4 - 4 = 0 \).

Step 2: Calculate the Change in x (Horizontal Change)

Identify the x-coordinates of the given points: the first point is (2,4) and the second point is (1,4). Calculate the difference in x-values: \( x_2 - x_1 = 1 - 2 = -1 \).

Step 3: Determine the Slope Using the Formula

Use the formula for the slope, which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the calculated vertical and horizontal changes: \( m = \frac{0}{-1} = 0 \).

Step 4: Classify the Line Based on the Slope

Since the slope of the line is 0, the line is horizontal. A line is classified as horizontal if it has a slope of 0.

Key Concepts

Horizontal Line

A horizontal line is a straight path that goes from left to right across a graph. It has no vertical movement, which means that all points on the line have the same y-coordinate. In coordinate geometry, this translates to a slope of 0. The slope, represented by the letter \( m \), measures the "steepness" of the line. For a horizontal line, this means there is no steepness because it doesn't rise or fall. Instead, it stretches endlessly along the same horizontal level. If you were to draw it or look at it, imagine a calm, flat horizon where the sky and land meet.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes a coordinate plane to investigate and understand geometric shapes. Within this framework, each point is defined by an ordered pair of numbers, \((x, y)\). These numbers represent the point's position along the x-axis (horizontal) and y-axis (vertical). By such representation, we can calculate distances, midpoints, and magnitudes like the slope. Coordinate geometry is crucial for finding the slope of a line, relating to how much a line ascends or descends across the plane. It transforms geometric relationships into algebraic formulas, making problem-solving more straightforward.

Calculating Slope

The slope of a line is a measure that describes the direction and steepness of the line. To calculate it, you subtract the y-coordinates and divide by the difference in x-coordinates, given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). When the difference in y-coordinates is zero, as in our original exercise, the slope becomes zero, indicating a horizontal line. If the line steepens upwards, the slope is positive, while a downwards slope is negative. Calculating the slope helps us not only determine direction but also classify the line.

Line Classification

Classifying lines involves determining their orientation based on the slope value. A line can be increasing, decreasing, horizontal, or vertical:

• Horizontal Line: The slope is 0. This line runs parallel to the x-axis.
• Vertical Line: The slope is undefined, as the change in x-coordinates \(x_2 - x_1 = 0\) causes division by zero. These lines run parallel to the y-axis.


• Increasing Line: The slope is positive. Such lines rise from left to right.
• Decreasing Line: The slope is negative. These lines fall from left to right.

Understanding these classifications helps in studying the behavior of lines in geometric problems.