Problem 32 - The Calculus with Analytic Geometry

Question

(a) Write an equation whose graph is the \(x\) axis. (b) Write an equation whose graph is the \(y\) axis. (c) Write an equation whose graph is the set of all points on either the \(x\) axis or the \(y\) axis.

Step-by-Step Solution

Answer

\[ y = 0 \], \[ x = 0 \], and \[ x = 0 \lor y = 0 \]

Step 1: Identify the equation for the x-axis

The x-axis is the horizontal line where the value of y is always 0, regardless of the x value. Therefore, the equation is: \[ y = 0 \]

Step 2: Identify the equation for the y-axis

The y-axis is the vertical line where the value of x is always 0, regardless of the y value. Therefore, the equation is: \[ x = 0 \]

Step 3: Identify the equation for points on either axis

For points to lie on either the x-axis or the y-axis, the points must satisfy either of the two previous equations. This can be represented using a logical OR: \[ x = 0 \lor y = 0 \]

Key Concepts

Understanding the x-axis

In coordinate geometry, the x-axis is a fundamental concept. It is the horizontal line on a graph that runs left to right. Every point on the x-axis has a y-coordinate of 0. This means that in an equation, if you see y = 0, it signifies the x-axis. The x-axis acts as a reference point for measuring vertical distances and is crucial for plotting points and graphing functions.

Getting to Know the y-axis

Similar to the x-axis, the y-axis is a crucial part of coordinate geometry. This vertical line runs up and down the graph. Points on the y-axis have an x-coordinate of 0. Therefore, the equation x = 0 represents the y-axis. The y-axis is used to measure horizontal distances and is essential for understanding the placement of points and functions on a graph.

Equations of Lines

Equations of lines are the backbone of coordinate geometry. They help us understand the position and slope of lines on a graph. For instance, the equation y = mx + c is the general form of a straight line, where m is the slope, and c is the y-intercept. However, for specific lines like the x-axis and y-axis, the equations are simpler. The x-axis has the equation y = 0, while the y-axis is represented by x = 0. Knowing these equations helps in plotting lines and points accurately on graphs.

Logical OR in Equations

The use of the logical OR in equations is essential when dealing with multiple conditions. In the context of coordinate geometry, to find the set of points that lie either on the x-axis or the y-axis, we use the logical OR. This is represented mathematically as x = 0 or y = 0. It means that a point can either have an x-coordinate of 0, aligning it with the y-axis, or a y-coordinate of 0, aligning it with the x-axis. Understanding this concept helps in identifying and solving more complex geometrical problems.