Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Give the equation of the \(x\) -axis.

Step-by-Step Solution

Answer

The equation of the x-axis is \( y = 0 \).

Step 1: Understanding the x-axis

The x-axis is the horizontal line on a coordinate plane. It is the line where all y-values are equal to zero. Therefore, any point on the x-axis can be written as \(x, 0\).

Step 2: Formulating the Equation

Since every point on the x-axis has a y-coordinate of zero, the equation that represents all these points involves y. The equation of a line where y is constant for all x-values is \(y = 0\). This captures the idea that no matter what the x-value is, the y-value remains zero on the x-axis.

Key Concepts

Coordinate plane

Imagine a large piece of graph paper with lines running both horizontally and vertically. This is much like what a coordinate plane looks like in mathematics. A coordinate plane consists of two main lines:

• The **x-axis**, running horizontally (left to right).
• The **y-axis**, running vertically (up and down).

They intersect at a point called the **origin**, which has the coordinates (0, 0).

On this grid, any location can be defined by a pair of numbers known as **coordinates**. The first number corresponds to the position along the x-axis, and the second corresponds to the position along the y-axis, written as (x, y). The coordinate plane is essential for graphing equations and understanding geometric concepts. It provides a visual way to solve mathematical problems and analyze relationships between numbers.

Horizontal line

A horizontal line is simply a straight line that runs from left to right and maintains the same height throughout its path. In the context of the coordinate plane, the x-axis itself is a great example of a horizontal line.

Key characteristics of horizontal lines include:

• **Constant y-coordinate**: Every point on a horizontal line has the same y-coordinate. This means it neither rises nor falls as you move along it.
• **Variable x-coordinate**: While the y-coordinate doesn't change, the x-coordinate can be any real number. Thus, points on a horizontal line might be (x, 3) or (x, -2), depending on the line's location on the coordinate plane.
• **Equation format**: The general form of a horizontal line is (y = k), where k is the y-coordinate for all points on that line. For the x-axis, since the y-coordinate is always zero, its equation is (y = 0).

Horizontal lines are simple yet powerful concepts in algebra and geometry, providing foundational knowledge for understanding graphs and linear equations.

Y-coordinate

The y-coordinate of a point is its "vertical" measure on the coordinate plane. It tells us how far up or down a point is from the x-axis. In any ordered pair (x, y), the y-coordinate is always the second number. Here's what you need to know:

• **Determines vertical position**: Larger y-coordinates mean the point is higher up, while smaller y-coordinates (including negative values) indicate it's lower down.
• **Equals zero on the x-axis**: For any point on the x-axis, the y-coordinate is zero because these points do not rise above or drop below this line.
• **Influences shaping of graphs**: The y-coordinate helps determine the shape and position of plotted graphs or lines on the coordinate plane.

Understanding how y-coordinates work is crucial for interpreting graph data, plotting functions, and solving equations involving vertical positioning on the plane.