Question

All lines are in the \((x, y)\) plane. Write, in parametric form, the equation of the \(y\) axis.

Short answer

The parametric equation of the y-axis is \[ \begin{cases} x = 0 \ y = t \end{cases} \].

Step-by-step solution

Identify the Characteristics of the y-Axis

The y-axis is a vertical line in the \((x, y)\) plane. This line passes through all points where the x-coordinate is zero.

Set the x-Coordinate to Zero

Since the y-axis includes all points where \(x = 0\), we can denote this as \(x = 0\) for all t.

Determine the y-Coordinate

The y-coordinate can vary freely along the length of the y-axis. Therefore, it can be represented as \(y = t\) where t is any real number.

Formulate the Parametric Equations

By combining the fixed x-coordinate and the variable y-coordinate, the parametric equations of the y-axis can be written as: \[ \begin{cases} x = 0 \ y = t \end{cases} \]

Check the Result

Verify that these parametric equations cover all the points on the y-axis. Any point on the y-axis has \(x = 0\) and can have any value for \(y\), so indeed these equations are correct.

Key concepts

y-axis

The y-axis is one of the fundamental components of the Cartesian coordinate system. It is a vertical line that divides the plane into two equal halves. This axis is characterized by having an x-coordinate of zero for all its points. In simple terms, any point on the y-axis can be represented as \( (0, y) \), where \(y\) can be any real number.
This essential feature makes it easy to understand and use in various mathematical problems.
Knowing how to write the equation of the y-axis in parametric form is crucial for solving problems related to coordinates and plotting.

vertical line in the plane

A vertical line in the plane is a line that runs straight up and down. Unlike horizontal lines, where the y-coordinate remains constant, vertical lines have a fixed x-coordinate. In a Cartesian coordinate system, this means you will often see equations like \( x = a \), where \(a\) is a constant.
For example, the y-axis itself is a vertical line described by \( x = 0 \). The fundamental property of vertical lines is that for any value of \(y\), the x-value remains unchanged, simplifying many types of calculations.
Understanding this concept is especially useful in graphing and solving geometric problems.

coordinate system

A coordinate system is a method for identifying unique positions on a plane using ordered pairs, commonly known as coordinates. In the most commonly used Cartesian coordinate system, each point is represented by a pair \( (x, y) \), which defines its location relative to two perpendicular lines called axes.
The horizontal line is the x-axis, while the vertical line is the y-axis. This system allows us to describe the position of points precisely and is widely used in mathematics, physics, engineering, and many other fields.
Knowing how to navigate the coordinate system is foundational to understanding more complex concepts like parametric equations and vertical lines.

parametric form

The parametric form of representing equations allows us to express the coordinates of the points on a curve as functions of one or more parameters. In the case of the y-axis, we use a single parameter, often denoted by \(t\).
The parametric equations for the y-axis are \begin{cases} x = 0 \ y = t \end{cases} , where \(t\) is a real number. This form provides a flexible way to describe geometric shapes and motions, making it invaluable in various applications like physics and computer graphics.
By learning how to transform standard equations into parametric form, students can tackle a wide array of mathematical and applied problems more efficiently.