Problem 23 - Algebra A Combined Function

Question

Write an equation of each line. See Examples 3 and \(4 .\) Undefined slope; through (0,5)

Step-by-Step Solution

Answer

The equation is \( x = 0 \).

Step 1: Understanding Undefined Slope

An undefined slope means the line is vertical. Vertical lines have the same x-coordinate for all points on the line.

Step 2: Identify the x-coordinate

Since the line is vertical and it passes through the point (0,5), the x-coordinate for all points on this line is 0.

Step 3: Write the Equation of the Vertical Line

The equation of a vertical line passing through a point with an x-coordinate of 0 is simply \( x = 0 \). This means no matter what the y-value is, the x-value remains constant at 0.

Key Concepts

Vertical Line

A vertical line is one of the fundamental concepts in coordinate geometry. Imagine a line that goes straight up and down on a grid. That's a vertical line! It stands perpendicular to the horizontal axis, often known as the x-axis.
Vertical lines are unique because, unlike other lines, they do not "slant" at all.These lines have a very special and unique characteristic: they have an undefined slope. Why is this the case? The slope of a line is a measure of its steepness, often described as "rise over run" or the change in y divided by the change in x (\( \frac{\Delta y}{\Delta x} \)). But, with vertical lines, there is no change in x! The line does not "run" horizontally; it moves vertically without moving to the left or right. Since we cannot divide by zero (the change in x is zero), the slope is undefined.

Equation of a Line

The equation of a line is a way to describe all the points that lay on that particular line. For most lines, this equation takes the form of the slope-intercept form, which is \[ y = mx + c \] where \(m\) is the slope and \(c\) is the y-intercept. However, when we deal with vertical lines, the equation is much simpler.
For a vertical line, every point on the line has the same x-coordinate. This means the equation of the line only needs to state this x-coordinate. Therefore, the equation of a vertical line is written as:

• \( x = a \)

where \(a\) is the constant x-value for all points on the line. No matter what y-value any point might have, the x-value remains the same.

x-coordinate

The x-coordinate is a fundamental part of any point on a graph. It represents a point’s position along the horizontal axis and is always the first number in an ordered pair (x, y). For vertical lines, however, the x-coordinate is a constant value for every point on the line.
Consider a vertical line that passes through the point (0,5). Here, the x-coordinate is 0, indicating that every point on this line will have an x-value of 0. This distinction is crucial in writing the equation for a vertical line, which, in this case, would simply be \( x = 0 \). The y-coordinate can vary infinitely along the vertical line, but the x-coordinate remains fixed.

Coordinate Geometry

Coordinate geometry, sometimes referred to as analytic geometry, allows us to describe geometric shapes using a coordinate system. In this system, any point can be defined using pairs of numbers, known as coordinates, in the form (x, y).
Vertical lines are a straightforward application of coordinate geometry. When a vertical line is plotted on a coordinate plane, it highlights the importance of keeping track of values on the x-axis. As described earlier, the equation \( x = a \) succinctly captures every point on a vertical line that shares the constant x-coordinate \(a\). Coordinate geometry aids in visualizing how different lines, including vertical ones, interact, allowing for a deeper understanding of geometrical principles. With this understanding, one can easily derive and interpret the equations of various types of lines.