Problem 60 - 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. $$ (-1,4) \text { and }(3,-1) $$

Step-by-Step Solution

Answer

The slope is \(-\frac{5}{4}\) and the line is decreasing.

Step 1: Identify Point Coordinates

First, we need to clearly identify the coordinates of the two points. The given points are (-1, 4) and (3, -1).Let \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (3, -1)\).

Step 2: Calculate the Slope

The slope \( m \) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points:\[m = \frac{-1 - 4}{3 - (-1)} = \frac{-5}{4}\]The slope of the line is \( -\frac{5}{4} \).

Step 3: Determine the Line Type Based on Slope

The nature of the line can be determined by the sign of the slope:- If the slope \( m > 0 \), the line is increasing.- If \( m < 0 \), the line is decreasing.- If \( m = 0 \), the line is horizontal.- If the slope is undefined/\( m \) involves division by zero, the line is vertical.Since \( m = -\frac{5}{4} < 0 \), the line is decreasing.

Key Concepts

Coordinate Geometry

Coordinate geometry is an essential part of mathematics that combines algebra and geometry. It allows us to place geometric figures in a coordinate plane and understand their properties through algebraic equations. This makes it easier to solve geometric problems using a coordinate system. In our problem, we use coordinate geometry to determine the slope of a line passing through two points on the plane.

• Coordinate Plane: A two-dimensional surface where each point is defined by a pair of numerical coordinates (for example, (x, y)).
• Points in Coordinate Geometry: Each point on the plane is represented by a set of coordinates.

Using coordinate geometry helps us visualize the problem and offers a systematic approach to calculate relationships, such as finding the slope of a line by using algebraic formulas.

Increasing and Decreasing Lines

When analyzing lines on a coordinate plane, understanding if they're increasing or decreasing is crucial. The slope of a line indicates this behavior.

• Increasing Lines: If the slope (\(m\) ) is positive, the line rises as it moves from left to right. This indicates that as the x-coordinate increases, the y-coordinate also increases.
• Decreasing Lines: Conversely, if \(m < 0\), the line descends from left to right. Thus, as the x-coordinate increases, the y-coordinate decreases.

These characteristics are essential for interpreting the behavior of lines in mathematical contexts and real-life scenarios such as in business to understand trends and change.

Horizontal and Vertical Lines

Lines that are either horizontal or vertical have unique characteristics.

• Horizontal Lines: These lines run parallel to the x-axis. The slope of a horizontal line is always 0 because the y-coordinates do not change as the x-coordinates vary (\(y_2 - y_1 = 0\)). Their equation is in the form \(y = c\), where \(c\) is a constant.
• Vertical Lines: Vertical lines run parallel to the y-axis, which means the x-coordinates remain constant (\(x_2 - x_1 = 0\)). Therefore, the slope is undefined because it involves division by zero. The equation of a vertical line is \(x = k\), where \(k\) is a constant.

Recognizing horizontal and vertical lines helps in solving complex geometrical problems and understanding graphs effectively.