Problem 32 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \((2, 4)\), \((4, -4)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((2,4)\) and \((4,-4)\) is -4.

Step 1: Plotting the Points

Plot both given points, which are \((2,4)\) and \((4,-4)\), on the Cartesian coordinate system. This will give a visual representation of the points and their position relative to each other.

Step 2: Identify the Coordinates

Identify and write down the coordinates of the two points. You have point 1 with coordinates \((x_1, y_1)\) as \((2, 4)\) and point 2 with coordinates \((x_2, y_2)\) as \((4, -4)\).

Step 3: Calculate the Slope

Substitute the values of the coordinates into the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Therefore \(m = \frac{-4 - 4}{4 - 2} = \frac{-8}{2} = -4\). We obtain a negative slope, which indicates that the line is declining.

Key Concepts

Understanding the Coordinate System

To begin with, let's dive into the basics of the coordinate system, a fundamental framework for graphing equations and plotting points. This system consists of two perpendicular lines, known as axes, which intersect at a point called the origin. The horizontal axis is labeled as the X-axis, while the vertical axis is the Y-axis.

In a 2D coordinate system, each point is determined by a pair of numerical coordinates. The first number, or the x-coordinate, indicates the point's horizontal position relative to the origin. The second number, or the y-coordinate, establishes the point's vertical position. Coordinates are usually written in parentheses \(x, y\).

A point's location is found by moving along the X-axis to the x-coordinate value, then vertically in line with the y-coordinate. Understanding how to navigate this system by plotting points and interpreting their positions is essential for graphing lines and shapes as well as for finding the slope.

The Slope Formula Explanation

The concept of the slope of a line is critical in understanding how to graph lines and analyze their steepness or flatness. Slope is essentially the measure of the steepness of a line. The slope formula is a way to quantify this concept and is written as \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(m\) represents the slope, \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are the coordinates of the second point.

The formula calculates the vertical change (difference in y-coordinates) divided by the horizontal change (difference in x-coordinates) between two points. A positive slope means the line rises from left to right, while a negative slope indicates the line falls. If the slope is zero, the line is horizontal and does not rise or fall. It's an invaluable tool in numerous areas, including calculus, economics, and physics.

Plotting Points Strategically

After understanding the coordinate system and slope, plotting points becomes a breeze. This process involves placing dots on the graph where the coordinates intersect. The strategy in plotting is simple: start at the origin, move along the x-axis to reach the x-coordinate, and then shift vertically to the y-coordinate.

For clarity, follow these steps:

• Locate the x-coordinate on the X-axis.
• Move directly up or down to the y-coordinate.
• Place a point where your line meets the Y-axis value.

This method provides a visual representation that can be used to see real-world data or to solve problems in mathematics, like finding the slope of a line. Consistent practice with plotting points helps in strengthening your understanding of the relation between algebraic equations and their graphic counterparts.