Question

Graph the line that contains the point P and has slope \(\mathrm{m}\). $$ P=(2,-4) ; m=0 $$

Short answer

The line is horizontal with the equation y = -4 passing through the point (2, -4).

Step-by-step solution

Understand the Given Information

Identify the given point and the slope. The point is P=(2, -4) and the slope is m=0.

Recall the Definition of a Horizontal Line

The slope m=0 indicates that the line is horizontal. A horizontal line has the same y-coordinate for all points on the line.

Determine the Equation of the Line

For a horizontal line passing through a point P=(2, -4), the equation of the line is y = -4. This is because every point on this horizontal line must have the same y-coordinate, which is that of the given point.

Plot the Line

To graph the line, plot the point (2, -4) on the coordinate plane. Then draw a horizontal line through this point. Every point on this line will have a y-coordinate of -4.

Key concepts

Coordinate Geometry

Coordinate Geometry involves plotting points, lines, and shapes on a coordinate plane. Each point on the plane is defined by an ordered pair \((x, y)\), which gives its position based on the x-axis (horizontal) and y-axis (vertical).
To graph a line, we typically need a point and a slope or the equation of the line. In our exercise, the given point is \(2, -4\).
When dealing with horizontal lines, the y-coordinate remains constant, which makes it simpler to plot.
For example, the point \(2, -4\) will be exactly at 2 units right of the origin on the x-axis, and 4 units down on the y-axis.

Slope and Intercept

Slope is a measure of how steep a line is. It's defined as the ratio of the rise (change in y) over run (change in x), represented by \( m \).
In this exercise, the slope \( m = 0 \) signifies that the line is horizontal, hence there is no vertical change as we move along the line.
This means that regardless of the x-value, the y-value remains the same. Intercept refers to the point where the line crosses the y-axis, known as the y-intercept.
For our horizontal line passing through the point \(2, -4\), the y-coordinate is always -4. Hence, the y-intercept is -4 and the equation of our line becomes \( y = -4 \).

Linear Equations

Linear Equations represent straight lines on the coordinate plane and are commonly written in the form \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept.
A special case of linear equations is horizontal lines where the slope \( m \) is zero. Thus, the equation simplifies to \( y = b \).
In this exercise, since the slope is 0 and the y-intercept given by point \(P = (2, -4)\) is -4, our equation is \( y = -4 \).
Graphing this, we plot the point \((2, -4)\), drawing a horizontal line through it, making sure all y-coordinates along the line are -4.