Problem 61 - 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. (3,5) and (-1,2)

Step-by-Step Solution

Answer

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

Step 1: Identify Coordinates

First, identify the coordinates of the given points. The points are (3,5) and (-1,2). Assign these as \((x_1, y_1) = (3, 5)\) and \((x_2, y_2) = (-1, 2)\).

Step 2: Apply Slope Formula

Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find the slope of the line passing through the points. Substitute the identified coordinates into the formula: \( m = \frac{2 - 5}{-1 - 3} = \frac{-3}{-4} \).

Step 3: Simplify the Fraction

Simplify the fraction obtained for the slope. The slope \( m = \frac{-3}{-4} = \frac{3}{4} \).

Step 4: Determine Line Type

Determine if the line is increasing, decreasing, horizontal, or vertical based on the slope value. Since the slope \( \frac{3}{4} \) is positive, the line is increasing.

Key Concepts

Coordinate Geometry

Coordinate Geometry stands as a branch of geometry where you use algebra with the coordinate plane. It combines geometric principles and algebraic equations by plotting points on a plane. In this plane, any point can be represented using ordered pairs known as coordinates—comprised of an x-coordinate (or abscissa) and a y-coordinate (or ordinate).

It's like giving an address to a point. For example, in the problem, the point (3,5) means you move three units along the x-axis and five units up on the y-axis. This foundational understanding helps you to visualize and analyze geometric situations, such as the location and distance of points.

• Coordinates are written as (x, y).
• The x-axis is horizontal, while the y-axis is vertical.
• Point coordinates indicate where the point is located on a grid.

Slope Calculation

Slope Calculation is critical for understanding the steepness and direction of a line on a graph. Slope represents how much the y-coordinate changes with respect to changes in the x-coordinate. The formula to calculate the slope (m) between two points (\(x_1, y_1\)) and \((x_2, y_2\)) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This fraction lets you gauge the change in vertical distance (rise) over the change in horizontal distance (run).

If the slope is positive, as with our example slope \(\frac{3}{4}\), moving from left to right, the line will rise.

• A positive slope indicates an increasing line.
• A negative slope suggests a decreasing line.
• Infinite or undefined slope is seen in vertical lines.
• Zero slope appears in horizontal lines.

By calculating slope, you gain insight into how quickly one variable changes concerning another. It is the essential step for further analysis of linear equations.

Linear Equations

Linear Equations describe a straight line using algebraic expressions. They are crucial in forming the equation of a line when the slope is known. Typically, one of the most common forms is the slope-intercept form:\[ y = mx + b \]Here, \(m\) represents the slope, and \(b\) stands for the y-intercept, which is where the line crosses the y-axis. This straightforward equation is handy for drawing lines and understanding how variables interact linearly.

Knowing the slope and a point, you can pinpoint the line's behavior and graph it accurately.

• The slope \(m\) dictates the line's direction and steepness.
• The y-intercept \(b\) shows the starting point on the y-axis.
• Every linear equation features a constant rate of change.

In everyday math problems, linear equations allow us to model real-life situations, such as calculating expenses over time or determining speed in motion problems, making them incredibly practical and relevant to both academic contexts and real-world applications.