Problem 16 - Thomas Calculus

Question

In Exercises 13–16, find an equation for (a) the vertical line and (b) the horizontal line through the given point. $$ (-\pi, 0) $$

Step-by-Step Solution

Answer

Vertical line: \(x = -\pi\); Horizontal line: \(y = 0\).

Step 1: Identify the Point

The given point is \((-\pi, 0)\), where \(-\pi\) is the x-coordinate and 0 is the y-coordinate.

Step 2: Write the Equation for a Vertical Line

The equation of a vertical line through a point \((a, b)\) is given by \(x = a\). Since our point is \((-\pi, 0)\), the equation for the vertical line is \(x = -\pi\).

Step 3: Write the Equation for a Horizontal Line

The equation of a horizontal line through a point \((a, b)\) is given by \(y = b\). Since our point is \((-\pi, 0)\), the equation for the horizontal line is \(y = 0\).

Key Concepts

Vertical Line

A vertical line is one that runs straight up and down the page, much like the direction of a skyscraper peering at the sky. The equation for a vertical line does not involve the y-coordinate at all. Instead, it's expressed using only the x-coordinate of the point it intersects. This is because a vertical line will forever intersect all infinite y-values along a single x-value.
To put it in simpler terms, think of a train track that only runs north and south. No matter where you are on the track, you will always have the same position in relation to east or west - this is similar to the x-coordinate in a vertical line.
The equation of a vertical line through any given point \(a,b\) is written as \(x = a\). It tells us that for all values of y, x will always be a constant. In the exercise example, with the point \((-\pi, 0)\), the vertical line equation is \(x = -\pi\). Every point on this line has an x-coordinate of \(-\pi\), making it consistent along its length.

Horizontal Line

A horizontal line is like a ruler lying flat on a table that goes from side to side, east to west in direction. Unlike vertical lines, horizontal lines have an unchanging y-coordinate and will stretch infinitely along different x-values. This feature becomes their defining characteristic.
The equation to describe a horizontal line is based on the fixed y-coordinate. If you were to walk along a completely flat plane, like a gym floor, your altitude (or height) doesn't change, regardless of your position. This constancy is represented by the y-coordinate remaining the same, no matter the x-value.
For a more mathematical view, the formula for a horizontal line that crosses through a point \(a,b\) is \(y = b\). For our particular point \((-\pi, 0)\), the horizontal line is described by \(y = 0\), meaning every single point on this line maintains a y-coordinate of 0. No matter the point's location left or right along the x-axis, its y-value will remain the same at 0.

Coordinates

Coordinates are fundamental to understanding how points are placed on a two-dimensional plane. When you refer to coordinates, you’re essentially talking about a specific spot on what could be seen as a large grid that spans infinitely.
Coordinates come in pairs, usually written as \(x, y\). The first number, x, tells you how far along the horizontal axis (side to side) the point is. The second number, y, indicates the point's position along the vertical axis (up and down).

• **X-coordinate:** Represents position relative to the vertical direction.
• **Y-coordinate:** Represents position relative to the horizontal direction.

They help in plotting points on graphs, finding solutions to equations, and analyzing graphs. In the case of our example point, \((-\pi, 0)\), this shows that the point is positioned \-\pi\ units from the origin along the x-axis, but isn't raised or lowered along the y-axis. Understanding coordinates is key to not just plotting points, but also understanding how lines like horizontal and vertical ones interact with the grid.