Problem 54 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (0,12), perpendicular to the \(x\) -axis

Step-by-Step Solution

Answer

The equation of the line is \( x = 0 \).

Step 1: Identify Characteristics of the Line

The line is described as perpendicular to the x-axis. A line perpendicular to the x-axis is a vertical line. Vertical lines have an undefined slope.

Step 2: Use the Given Point

The line passes through the point (0,12). For vertical lines, the x-coordinate remains constant. Therefore, the equation of the line will be of the form \( x = a \), where \( a \) is the x-coordinate of the given point.

Step 3: Write the Equation of the Line

Since the line is vertical and passes through (0,12), the equation is simply \( x = 0 \). Vertical lines cannot be expressed in slope-intercept form, \( y = mx + b \), because their slope is undefined.

Key Concepts

Vertical Lines

A vertical line is a special type of line in geometry. It runs straight up and down and is perfectly perpendicular to the x-axis on a Cartesian plane.
Vertical lines have a distinct property from other lines, which is that they keep the x-coordinate constant while the y-coordinate can be any value. For example, if a vertical line passes through the point (3, y), it means that x=3 for any y value.
Vertical lines help describe situations where there is a constant x-value across all y-values. This characteristic makes them unique since they are not described by the usual slope-intercept form of a line equation, which is used for non-vertical lines.

Undefined Slope

In mathematics, the slope of a line indicates its steepness and direction. However, the slope of a vertical line is referred to as undefined. This is due to the fact that the change in x (\(\Delta x \)) is zero, as the x-value remains the same, while there is a change in y (\(\Delta y \)).
The formula for calculating slope is given by \( m = \frac{\Delta y}{\Delta x} \). For a vertical line, this results in division by zero, which is undefined in mathematics. Thus, we say that the slope of a vertical line is undefined.
It’s crucial to remember that when dealing with vertical lines, the concept of slope does not apply in the same way it does to other lines. Understanding this helps avoid confusion, especially when attempting to fit a vertical line into the slope-intercept form.

Equation of a Line

The equation of a line in geometry is a mathematical way to represent it on a graph. For most lines, the familiar form used is the slope-intercept form given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
However, for vertical lines, this form does not apply because their slope is undefined. Instead, vertical lines are represented by equations that state the constant x-value. Such an equation looks like \( x = a \), where \( a \) is the x-coordinate of any point on the line.
This form clearly states that the line includes all y-values for a specific x-value. It’s simple and effective for describing vertical lines, involving direct observation of the line’s placement on the graph. Knowing how to express the equation of both vertical and non-vertical lines is fundamental in geometry and graphing.