Problem 27 - Elementary Algebra

Question

For the following problems, graph the equations. $$ y=-2 $$

Step-by-Step Solution

Answer

Answer: The equation y = -2 represents a horizontal line when graphed, and it crosses the y-axis at the point (0, -2).

Step 1: Identify the equation type

We have the equation y = -2, which is a horizontal line because there is no x variable present in the equation.

Step 2: Determine the y-intercept

The y-intercept is the point where the line crosses the y-axis. In this case, it's given directly by the equation: y = -2. The line will cross the y-axis at the point (0, -2).

Step 3: Find points on the line

Since this is a horizontal line, there are infinitely many points with y = -2. We can choose a few points to better visualize the line. Some examples of points on the line include (0, -2), (1, -2), (-1, -2), (2, -2), and (-2, -2).

Step 4: Draw the line

Using the points that we found in step 3, draw a horizontal line that goes through all of these points. The line should be parallel to the x-axis and have a y-coordinate of -2 throughout its length. This line represents the graph of the equation y = -2.

Key Concepts

Horizontal Line Equation

When we talk about the horizontal line equation, we are referring to a line that runs left to right across the coordinate plane. Unlike other linear equations that can have varying slopes and intercepts, a horizontal line has a very specific characteristic: its slope is 0. This means that no matter how far along the x-axis you go, the y-value remains constant.

For example, the equation from our exercise, \( y = -2 \), indicates that the line has a y-value of -2 at every point. This equation is as simple as it gets when dealing with linear graphing: there's no x variable involved. To plot this line on a coordinate plane, one must simply draw a straight, horizontal line through the entire grid at the y-coordinate of -2. Since it never goes up or down, we don't have to worry about calculating a slope, making it one of the easiest types of linear equations to graph.

Y-Intercept

The y-intercept is another critical concept to understand when graphing linear equations. It is the point where the line crosses the y-axis. In algebra, the y-intercept is often written as \( (0, b) \), where \( b \) is the value of the y-coordinate at the point where the line intersects the y-axis.

The exercise presents us with the equation \( y = -2 \), which tells us directly that the y-intercept is \( (0, -2) \), meaning that the line crosses the y-axis at 2 units below the origin. When drawing any linear equation, finding the y-intercept gives you an important starting point. In the case of a horizontal line, the y-intercept is particularly straightforward, as it also represents every single point on the line.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is the two-dimensional space where we graph equations. It's made up of a horizontal number line called the x-axis and a vertical number line called the y-axis. The point at which these axes intersect is known as the origin, with coordinates \( (0,0) \).

In the coordinate plane, any point can be represented by an ordered pair of numbers, known as coordinates. The first number indicates the position along the x-axis, and the second number indicates the position along the y-axis. For our exercise, points like \( (0, -2) \), \( (1, -2) \), and \( (-1, -2) \) were chosen to illustrate the horizontal line \( y = -2 \). Understanding the layout and how to navigate the coordinate plane are fundamental skills in graphing, as they allow you to visually represent the relationship between numbers or variables.