Problem 64 - 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. . (1,9) and (-8,5)

Step-by-Step Solution

Answer

The slope is \(\frac{4}{9}\); the line is increasing.

Step 1: Identify Points and Formula

We are given the points (1, 9) and (-8, 5). To find the slope of a line passing through two points, we use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1) = (1, 9)\) and \((x_2, y_2) = (-8, 5)\).

Step 2: Substitute Values into Formula

Plug the values into the slope formula: \( m = \frac{5 - 9}{-8 - 1} \). This gives us the fractions \(\frac{-4}{-9}\).

Step 3: Calculate the Slope

Simplify \( \frac{-4}{-9} \). The slope is \( \frac{4}{9} \) because a negative divided by a negative results in a positive.

Step 4: Determine Line's Direction

Since the slope \(\frac{4}{9}\) is positive, this means the line is increasing. An increasing line goes upwards as it goes from left to right.

Key Concepts

Understanding an Increasing Line

An increasing line is one that rises as it moves from left to right on the coordinate plane. This happens when the slope of the line is positive. In simpler terms, if you imagine walking along the path of the line from left to right, you would be going uphill. This is a key concept in determining the nature of a line in graph-based mathematics. When you see a positive slope, it's an indicator that for every step to the right you take (an increase in x-value), you're also moving upwards (an increase in y-value).

• The larger the slope, the steeper the line.
• A perfectly flat line (slope of zero) is neither increasing nor decreasing.
• If the slope is negative, the line would be decreasing, heading downhill from left to right.

Understanding whether a line is increasing or decreasing helps in analyzing graphs and predicting trends in data.

Decoding the Slope Formula

The slope formula is a fundamental tool in coordinate geometry used to calculate a line’s steepness. It is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points on the line,
• \( y_2 - y_1 \) represents the change in the vertical direction (rise), and
• \( x_2 - x_1 \) represents the change in the horizontal direction (run).

Essentially, the slope formula divides the change in y (vertical change) by the change in x (horizontal change). This gives a ratio that defines the line's direction and steepness. The simplicity of this formula lies in its ability to quickly tell whether a line is ascending or descending, as well as how "tilted" it is. In our example, substituting into the formula gives us a positive slope of \( \frac{4}{9} \), confirming the line is increasing.

Exploring Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebra and the coordinate plane to study geometrical figures. The main elements are points, lines, and curves plotted on an x-y coordinate system. This form of geometry allows you to represent shapes numerically which simplifies complex geometric problems and makes it easier to visualize relationships between different figures.

• Points are defined by their coordinates \((x, y)\).
• Lines can be described using equations derived from the slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
• Curves are represented by equations like \(y = ax^2 + bx + c\) for parabolas.

Through coordinate geometry, you can find distances between points, midpoints, slopes, and even the area bounded by shapes on the plane. It's a crucial part of understanding and solving real-world problems by bridging algebra and geometry.