Question

Graph the line that contains the point P and has slope \(\mathrm{m}\). $$ P=(-2,0) ; \text { slope undefined } $$

Short answer

The line is vertical with the equation \(x = -2\).

Step-by-step solution

Identify the Given Information

The point given is \((-2, 0)\) and the slope of the line is undefined.

Understand What an Undefined Slope Means

A line with an undefined slope is a vertical line. Vertical lines have the same x-coordinate for all points on the line.

Determine the Equation of the Vertical Line

Since the line passes through the point \((-2, 0)\) and is vertical, all points on the line will have \(x = -2\). Therefore, the equation of the line is \(x = -2\).

Graph the Line

To graph the line, draw a vertical line that passes through the x-coordinate \(-2\) on the Cartesian plane.

Key concepts

undefined slope

When we talk about an undefined slope, we are referring to a situation where the slope of a line cannot be calculated. In mathematical terms, the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \((y_2 - y_1) / (x_2 - x_1)\).

If we find ourselves dividing by zero, which happens when \(x_1 = x_2\), the slope is undefined.

This occurs because the points form a vertical line that goes up and down. Thus, the line's slope cannot be measured in the usual way because the change in x is zero, leading to division by zero, an undefined operation.

vertical line

A vertical line is a straight line that moves up and down on the graph without any horizontal movement.

On the Cartesian plane, every point on a vertical line has the same x-coordinate. For instance, in our exercise, the point given is \((-2, 0)\) and the slope is undefined, meaning the line is vertical.
This denotes that every point on this line will have x = -2.

This type of line is characterized by the fact that it does not cross the y-axis at more than one point and is parallel to the y-axis.

equation of a line

In mathematics, the equation of a line provides a way to describe all the points that lie on that line. For a vertical line, the equation is simple.

If a vertical line passes through the point \((a, b)\), the equation of that line is \(x = a\).

For the given exercise, since the line passes through the point \((-2, 0)\), the equation is \(x = -2\).
This indicates that regardless of the y-coordinate, the x-coordinate will always be -2, visually forming a vertical line on the graph.

Cartesian plane

The Cartesian plane is a two-dimensional plane defined by a horizontal line called the x-axis and a vertical line called the y-axis.

These axes intersect at a point called the origin, denoted as (0,0).

Each point on the plane can be represented by a pair of coordinates \((x, y)\), where x represents the horizontal position and y represents the vertical position.
In the given exercise, the point \((-2, 0)\) indicates that the point is 2 units to the left of the origin and exactly on the x-axis.

By understanding these fundamental concepts, you can easily plot points and graph lines on the Cartesian plane.