Problem 59 - 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. $$ (-2,4) \text { and }(1,1) $$

Step-by-Step Solution

Answer

The slope is -1, and the line is decreasing.

Step 1: Identify the Points

Identify the pair of given points. The given points are \((-2, 4)\) and \((1, 1)\).

Step 2: Use the Slope Formula

To find the slope \(m\) of the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Substituting the coordinates \((x_1, y_1) = (-2, 4)\) and \((x_2, y_2) = (1, 1)\), we get: \[m = \frac{1 - 4}{1 - (-2)}\]

Step 3: Simplify the Slope Calculation

Perform the calculations to find the slope: \[m = \frac{-3}{1 + 2} = \frac{-3}{3} = -1\] Therefore, the slope of the line is \(-1\).

Step 4: Determine the Line's Characteristic

Based on the slope value: - If \(m > 0\), the line is increasing. - If \(m < 0\), the line is decreasing. - If \(m = 0\), the line is horizontal. - If the slope is undefined, the line is vertical. Since \(m = -1\) for our line, which is less than 0, the line is decreasing.

Key Concepts

Slope Formula

The slope of a line provides us with the steepness or direction of the line on a coordinate plane. The slope is defined as the change in the vertical direction, divided by the change in the horizontal direction.
In mathematical terms, for any two points i.e., \((x_1, y_1)\) and\((x_2, y_2)\), the slope \(m\) is calculated using the slope formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula is straightforward and involves subtracting the y-coordinates and x-coordinates of the points to find their respective differences.
Then, it requires dividing the difference in y-coordinates by the difference in x-coordinates.

• When the denominator equals zero (i.e., \(x_2 = x_1\)), the slope is undefined, indicating a vertical line.
• However, if the numerator equals zero (i.e., \(y_2 = y_1\)), the slope is zero, resulting in a horizontal line.

Using this formula helps in efficiently determining the line's steepness and direction, which are crucial in understanding line behavior in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to study geometrical shapes and figures using coordinates. Points, lines, and shapes are described within a coordinate plane, typically the Cartesian coordinate system.
In this system, each point is represented by a pair of numbers \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.
Lines, key to coordinate geometry, can be represented by using the slope and a point on the line, forming the equation of a line.

• Lines are often described by the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• The midpoint and distance between points can also be calculated using specific formulas integrated within this geometric framework.

These concepts play a vital role in higher-level geometry and calculus, providing the foundational understanding necessary to solve complex geometrical problems by converting them into algebraic tasks.

Line Characteristics

Understanding the characteristics of a line involves knowing its slope, which dictates the line's direction and rate of change. The slope, symbolized as \(m\), provides insight into whether a line rises, falls, runs horizontally, or stands vertically.

• A positive slope \(m > 0\) implies a line that ascends from left to right, indicating it is an increasing line.


• A negative slope \(m < 0\) suggests a line that descends from left to right, categorizing it as a decreasing line.


• A slope of zero \(m = 0\) indicates a perfectly horizontal line with no vertical change.


• An undefined slope occurs with vertical lines, where the change in x-coordinates is zero, making the slope formula's denominator zero.

These elements help us visualize and predict how lines traverse the coordinate plane, essential for analyzing graphs and solving real-world problems where direction and steepness of change are crucial.