Problem 3 - Introductory Algebra for College Students

Question

In Exercises \(1-10,\) find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(-2,1) \text { and }(2,2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-2,1) and (2,2) is \(\frac{1}{4}\). Therefore, the line rises.

Step 1: Calculate the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \(slope = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points (-2,1) and (2,2) into this formula, we get: \(slope = \frac{2 - 1}{2 - (-2)} = \frac{1}{4}\).

Step 2: Analyze the Slope to Determine the Orientation of the Line

The calculated slope is positive, hence the line rises.

Key Concepts

Calculating Slope

Understanding the slope of a line is fundamental in geometry and algebra. It dictates how steep the line is, which in turn, represents the rate at which one variable changes with respect to another. To calculate the slope, also known as the gradient, we take two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \), and apply the formula \( \frac{y_2 - y_1}{x_2 - x_1} \). This formula gives us the vertical change divided by the horizontal change between the two points.

For example, with the points \( (-2,1) \) and \( (2,2) \) from our exercise, we substitute these into our formula to find the slope: \( \frac{2 - 1}{2 - (-2)} = \frac{1}{4} \). A positive outcome like this indicates that for every four units we move to the right (horizontally), the line goes up by one unit (vertically). Thus, the line has a gentle incline.

Undefined Slope

Sometimes in mathematics, we encounter a special scenario when trying to calculate the slope of a line: a vertical line. Vertical lines have an 'undefined' slope because their formula involves division by zero, which is not possible in standard arithmetic. The reason behind this is that vertical lines have no horizontal change between any two points (\( x_1 = x_2 \)), so the denominator in our slope formula becomes zero.

In such cases, the formula \( \frac{y_2 - y_1}{x_2 - x_1} \) does not yield a numeral value, leading us to declare the slope as 'undefined’. This is an essential concept to remember because assuming any value other than 'undefined' can result in errors in calculations and misinterpretations of the line's behavior.

Line Orientation

The slope of a line also informs us about its orientation—whether it rises, falls, remains horizontal, or is vertical. When the slope is positive, as in our working example \( \frac{1}{4} \), the line rises, indicating an increasing function as you move from left to right. Conversely, a negative slope indicates a falling line, representing a decreasing function.

If the slope equals zero, the line is horizontal, showing no vertical change regardless of the horizontal movement; it represents a constant function. Lastly, for undefined slopes, where the line is vertical, it indicates that the function's value at a given \( x \) is not unique, leading to infinite possibilities for \( y \) at that \( x \) value. Recognizing these orientations helps in visualizing the graph of equations and understanding the relationship between variables.