Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Vertical and passing through }\left(-\sqrt{\pi}, \pi^{2}\right) $$

Step-by-Step Solution

Answer

The equation of the line is \(x = -\sqrt{\pi}\).

Step 1: Identify the x-coordinate

The point given is \(-\sqrt{\pi}, \pi^{2}\). The x-coordinate of this point is \(-\sqrt{\pi}\).

Step 2: Formulate the equation

Since the line is vertical, its equation will be in the form \(x = k\). Substituting the x-coordinate of the point for \(k\), the equation of the line is \(x = -\sqrt{\pi}\).

Key Concepts

Coordinates in the Cartesian Plane

When we talk about plotting points, we are dealing with a Cartesian plane, which is a two-dimensional space defined by an x-axis (horizontal) and a y-axis (vertical). The location of any point within this plane is given by a pair of numbers called coordinates. These are written in the form \( (x, y) \) and represent the position along the x-axis and the y-axis, respectively. For example, the point \( (-\sqrt{\pi}, \pi^{2}) \) refers to a point that is located at \( -\sqrt{\pi} \) units along the x-axis and \pi^{2}\ units up the y-axis.

Understanding how to plot and read these coordinates is fundamental when it comes to representing and finding the equations of lines because every line is essentially a collection of points that abide by a particular relationship between their x and y values.

Vertical Line Properties

Vertical lines in the Cartesian plane are unique in that they have an undefined slope. This is because these lines go straight up and down, parallel to the y-axis. One way to identify a vertical line is that all points on this line have the same x-coordinate. It doesn't matter how far up or down the line you go; the x-coordinate remains constant while the y-coordinate can be any value. This inherent property of the vertical line is what we use to formulate its equation.

This distinguishing characteristic means that the vertical line equation doesn't feature a y variable—unlike non-vertical lines, which are defined by equations that show a relationship between x and y. Consequently, understanding that a vertical line has a constant x-value is crucial when determining its equation.

Equation Formulation for Lines

The general equation of a line in the Cartesian plane can take many forms, depending on the orientation of the line and the information available. A non-vertical line is typically expressed in the slope-intercept form: \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. For vertical lines, as we've established, there is no slope-intercept form due to the undefined slope. Instead, it takes the simple and distinct form \( x = k \) where \( k \) is the constant x-value for all points on the line.

When you're given a point through which a vertical line passes, like \( (-\sqrt{\pi}, \pi^{2}) \), you set \( k \) to the x-coordinate of that point to get the vertical line's equation. In our exercise, this gives us \( x = -\sqrt{\pi} \) as the equation for the line that passes vertically through the point \( (-\sqrt{\pi}, \pi^{2}) \) on the Cartesian plane. Emphasizing the consistent x-coordinate in this format is key to ensuring a proper understanding of vertical lines.