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Chapter 7: Problem 44

Graph each horizontal or vertical line. \(y=-3\)

Short Answer

Expert verified
The line \(y = -3\) is a horizontal line running through the y-coordinate of -3. This line includes all points where the y-value is -3, regardless of the x-value.

Step by step solution

01

Understanding the problem

The given function, \(y = -3\), is the equation of a line, specifically a horizontal line that intersects the y-axis at \(y=-3\). Since there are no stipulations on the x variable, it means that for any possible value of x, the value of y will remain -3.
02

Plotting the line

The way to plot this line is by selecting values for \(x\) and plotting those points with \(y = -3\). For instance, if you choose -2, 0, and 2 as values for \(x\), corresponding points on the line would be \((-2, -3)\), \((0, -3)\) and \((2, -3)\). All these points lie on the line \(y = -3\).
03

Drawing the graph

Now, the selected points can be plotted on a graph. Then the points must be connected with a straight line since this is a linear function. Remember: the line extends indefinitely in both directions, so arrows must be drawn at either ends.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate System
The coordinate system, also known as the Cartesian coordinate system, serves as the backbone of most graphing activities. It consists of two axes: the horizontal x-axis and the vertical y-axis, which intersect at a point called the origin (0,0).

Each point on the plane is defined by an ordered pair \(x, y\), where \(x\) is the position along the horizontal axis and \(y\) is the position along the vertical axis. For graphing horizontal lines like in the given exercise \(y = -3\), the y-coordinate remains constant and the x-coordinate can be any value, indicating that all points lie on the same horizontal level.
Linear Equations
Linear equations describe straight lines and have the general form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. In contrast, a horizontal line like \(y = -3\) has a slope \(m\) of 0. This indicates no rise over the run, meaning the line is flat and parallel to the x-axis.

A slope of zero is significant because it tells us that no matter how far we move along the x-axis, the y-value of the line does not change, resulting in a horizontal line. This property simplifies the process of graphing since we're dealing with a constant y-value for all x-coordinates.
Plotting Points
Plotting points is a fundamental step in graphing any equation. To accurately plot a horizontal line like \(y = -3\), choose a variety of x-values. Be sure to select a range that includes both negative and positive values to represent the line fully.

For each chosen x-value, pair it with the constant y-value given by the equation. In our example, points such as \(\(-2, -3\)\), \(\(0, -3\)\), and \(\(2, -3\)\) would be plotted. Remember to ensure that your points are accurately placed on the graph to reflect the true position of the line. After plotting the points, draw a straight line through them, extending to show the line continues indefinitely in both directions.
Y-Intercept
The y-intercept is where a line crosses the y-axis, and it's an essential feature of linear equations. For the equation of a horizontal line like \(y = -3\), the y-intercept is simply -3. This value indicates the single point on the line that touches the y-axis.

In a typical linear equation \(y = mx + b\), \(b\) represents the y-intercept. However, for horizontal lines, the entire equation reflects the y-intercept due to the slope being zero. This makes graphing horizontal lines straightforward since the line will intersect the y-axis at \(\(0, y\)\) where \(y\) is the constant value from the equation, which is -3 in our exercise.

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Most popular questions from this chapter

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