Problem 38 - Brief Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Horizontal and passing through the point \(\left(\frac{1}{2}, \frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the horizontal line is \(y = \frac{3}{4}\).

Step 1: Understand the Characteristics of a Horizontal Line

In a horizontal line, every point on the line shares the same y-coordinate. The line has a slope of 0.

Step 2: Identify the y-Coordinate

Since the horizontal line passes through the point \(\left(\frac{1}{2}, \frac{3}{4}\right)\), it shares the same y-coordinate, which is \(\frac{3}{4}\).

Step 3: Construct the Equation

For a horizontal line, the equation is simply \(y = c\), where \(c\) is the constant y-coordinate. So the equation is \(y = \frac{3}{4}\).

Key Concepts

Horizontal Lines

Horizontal lines are a significant concept in coordinate geometry. They play an important role in understanding graph behaviors and relationships between points. A horizontal line runs left to right parallel to the x-axis on a coordinate plane.

• All points on this line share the same y-coordinate value. This means no matter where you are on the line, the height (the y-value) remains constant.
• As a result, no vertical change occurs between any two points on a horizontal line.

Horizontal lines have a distinct characteristic: the slope is always zero. This zero slope indicates a flat surface, with no incline or decline.
Whether you come across a horizontal line in algebra or coordinate geometry, it will always take the form of an equation like \( y = c \), where \( c \) is the y-coordinate at all positions on the line.

Slope

Slope is a fundamental concept in algebra and geometry, indicating the steepness or incline of a line. Slope is calculated using "rise over run," which compares the vertical change to the horizontal change between two points on a line. In mathematical terms, it's defined as:

• \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\)

For a horizontal line, however, the slope is 0.

• This is because the *rise* between any two points is zero: all points on the line have the same y-coordinate.
• Thus, regardless of the horizontal distance (the "run"), the line does not incline or decline.

When identifying or writing equations, understanding the slope assists in predicting how the line behaves. In cases like our horizontal line, knowing the slope is zero aids in constructing a straightforward equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where algebra is used to describe geometric principles on a coordinate plane. It connects algebraic equations with geometric figures. Here's how it works:

• Points are represented as coordinates \((x, y)\), depicting positions on a two-dimensional plane.
• Each equation corresponds to a line or curve on this plane.

In the context of our problem, coordinate geometry helps us understand how a point like \(\left(\frac{1}{2}, \frac{3}{4}\right)\) relates to other points and lines.

• This integration of algebra and geometry allows us to derive precise equations for lines. For instance, horizontal lines through any point can be expressed simply due to their constant y-coordinate.
• The form \(y = c\) offers an uncomplicated representation of horizontal lines on this coordinate plane, illustrating how coordinate geometry simplifies complex problems into manageable parts.

By utilizing these principles of coordinate geometry, solving equations and understanding their graphical representations becomes more intuitive and accessible.