Problem 60 - Calculus Volume 1

Question

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) and (3,-1)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Slope Formula

To find the slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the formula: \\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]\Here, the given points are \( (-1, 4) \) and \( (3, -1) \). Let's apply these coordinates to the formula.

Step 2: Calculating the Numerator

Substitute the given \(y \) values into the formula: \\[ y_2 - y_1 = -1 - 4 \]\Calculate the expression: \(-1 - 4 = -5\).

Step 3: Calculating the Denominator

Substitute the \(x \) values into the formula: \\[ x_2 - x_1 = 3 - (-1) \]\Simplify by adding the opposite: \(3 + 1 = 4\).

Step 4: Finding the Slope

Now apply the results into the slope formula: \\[ m = \frac{-5}{4} \]\The slope of the line through the points \( (-1, 4) \) and \( (3, -1) \) is \( \frac{-5}{4} \).

Step 5: Classifying the Line

Since the slope \(m = \frac{-5}{4}\) is a negative number, the line is decreasing. This means it moves down as it goes from left to right.

Key Concepts

Slope Formula

The slope formula is a fundamental tool in algebra that helps us understand the steepness and direction of a line on a coordinate plane.
To compute the slope given two points, \((x_1, y_1)\) and \((x_2, y_2)\), we use the following formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

The slope is represented by \(m\), and this formula calculates the change in y (vertical change) over the change in x (horizontal change).
This can also be described as "rise over run," which reflects how much the line rises or falls as it moves horizontally. To apply the slope formula correctly:

• Identify the coordinates of the two points.
• Substitute the y-values into the numerator and the x-values into the denominator.
• Perform the arithmetic operations to simplify both the numerator and the denominator.

The slope formula is a quick way to gauge the line's direction and intensity.

Decreasing Line

A decreasing line is a type of linear graph with a negative slope. This means that as you move from left to right across the graph, the line goes downwards.
It signifies that there is an inverse relationship between the x-values and y-values. In simpler terms, when x increases, y decreases.Recognizing a Decreasing Line:

• If the slope \(m\) is negative, the line is decreasing.
• This can be visualized when plotting the line: the line will point downhill.

For example, in our previous exercise, the slope calculated was \( \frac{-5}{4} \).
This negative fraction clearly indicates a decreasing line. Understanding a line's increasing or decreasing nature is crucial for predicting behavior within a dataset or a function's graph.

Linear Equations

Linear equations are equations of the first order, which means they graph as a straight line on the coordinate plane. The general form of a linear equation is \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Linear equations can model a variety of real-world situations due to their simple and predictable nature.In practical applications:

• If the slope \(m\) is positive, the line increases.
• If \(m\) is negative, the line decreases.
• The y-intercept \(b\) indicates the value of y when x equals zero.

Using linear equations, you can visually represent relationships between variables and use this to make predictions or identify trends. This makes understanding and applying linear equations a powerful tool in both academic and real-world contexts.