Problem 61 - 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. $$ (3,5) \text { and }(-1,2) $$

Step-by-Step Solution

Answer

The slope is \(\frac{3}{4}\) and the line is increasing.

Step 1: Identify the Points

We are given two points, \((3, 5)\) and \((-1, 2)\). Let's refer to the first point as \((x_1, y_1)\) where \(x_1 = 3\) and \(y_1 = 5\), and the second point as \((x_2, y_2)\) where \(x_2 = -1\) and \(y_2 = 2\).

Step 2: Use the Slope Formula

The formula to find the slope \(m\) between two points is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given coordinates into the formula, we have:\[m = \frac{2 - 5}{-1 - 3}\]

Step 3: Calculate the Slope

Compute the difference in the \(y\) values: \(2 - 5 = -3\). Compute the difference in the \(x\) values: \(-1 - 3 = -4\). Therefore, the slope \(m\) is:\[m = \frac{-3}{-4} = \frac{3}{4}\]

Step 4: Determine the Line's Direction

Since the slope \(\frac{3}{4}\) is positive, the line is increasing. A positive slope indicates that as \(x\) increases, \(y\) also increases.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. Points on the plane are defined using pairs of numbers called coordinates, which represent the horizontal (x-axis) and vertical (y-axis) distances from a central point known as the origin.

The basic element of coordinate geometry is the point, denoted as

• The Coordinates: Written as <(x, y)>, where is the horizontal position and is the vertical position.

This system allows us to plot points, draw shapes, and even define lines and curves. Understanding the layout and positioning on the coordinate plane helps in solving problems related to distance, midpoint, and sections of lines, and is fundamental in graphing equations.

Slope Formula

The slope of a line is a number that describes its steepness and direction. It is calculated using the slope formula:

• Using Coordinates: Given two points <((x_1, y_1))> and <((x_2, y_2))>, the slope formula is:
  \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

The slope quantifies how much changes for a change in . It is a measure of the vertical change (rise), over the horizontal change (run). Calculating the slope is straightforward once you accurately identify the coordinates.
A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. Understanding how to apply the slope formula allows us to analyze the behavior of linear equations graphically.

Increasing and Decreasing Lines

In coordinate geometry, understanding whether a line is increasing, decreasing, horizontal, or vertical is crucial. This behavior is dictated by the slope value:

• Positive Slope ( > 0): Represents an increasing line. As the value increases, so does the value. This gives a line that rises when moving from left to right.
• Negative Slope (m < 0): Represents a decreasing line. The coordinate decreases with increasing , resulting in a line that slopes downwards.
• Zero Slope (m = 0): Produces a horizontal line. Here, the coordinate remains constant as varies, indicating no rise, just run.
• Undefined Slope: Found in vertical lines. Since there is no run, the division calculation is <\(\frac{0}{0}\)> causing undefined slope.

Recognizing these distinctions enables better graphical interpretations and predictions about the line's orientation and behavior on a coordinate plane.

Linear Equations

Linear equations form straight lines when graphed on the coordinate plane and are generally expressed in the form:

• Standard Form: \(Ax + By = C\), where , , and are constants.
• Slope-Intercept Form: \(y = mx + b\), where represents the slope and the y-intercept.

Linear equations exhibit constant rates of change characterized by their slope.
Solving linear equations involves finding the x and y intercepts, or using techniques like substitution or elimination in systems of linear equations. They can model real-world situations where there’s a proportional relationship between two quantities. Understanding their graphical representation helps in visual insights, allowing us to easily discern trends and make predictions.