Problem 83 - Intermediate Algebra

Question

Graph each equation. $$y=3$$

Step-by-Step Solution

Answer

The graph is a horizontal line at \(y=3\).

Step 1: Understanding the Equation

The equation given is \(y = 3\). This is a horizontal line because \(y\) is always equal to 3, regardless of the value of \(x\).

Step 2: Plotting Points

To graph the line, first plot a few points where \(y = 3\). For instance, when \(x = 0\), then \(y = 3\). Similarly, when \(x = 2\), \(y = 3\), and when \(x = -3\), \(y = 3\).

Step 3: Drawing the Line

After plotting the points (0,3), (2,3), and (-3,3), draw a straight horizontal line through these points. This line should extend infinitely in both horizontal directions, passing through the \(y\)-axis at \(y = 3\).

Step 4: Final Graph

The graph is a straight, horizontal line across the plane at the level \(y = 3\). Make sure the line is parallel to the \(x\)-axis and cuts the \(y\)-axis at the point (0,3).

Key Concepts

horizontal lines

In the world of graphing linear equations, horizontal lines play a special role. They are unique because they are perfectly parallel to the x-axis, making them incredibly simple to understand and graph. A horizontal line is represented by an equation of the form \( y = c \), where \( c \) is a constant. This means that the value of \( y \) remains the same, no matter what value \( x \) takes.
For example, in the equation \( y = 3 \), \( y \) will always be 3, regardless of whether \( x \) is 0, 101, or -55. This consistent value is what makes the line horizontal. Since every point on this line has the same y-coordinate, it results in a line that stretches infinitely on the coordinate plane left to right. This consistent nature makes plotting them on a graph both straightforward and reliable.

coordinate plane

The coordinate plane is a fundamental concept in graphing. It is a two-dimensional space formed by two axes: the horizontal \( x \)-axis and the vertical \( y \)-axis. These axes intersect at a point called the origin, denoted by (0, 0). The coordinate plane is divided into four quadrants which help in identifying the location of points.
In this structured space, every point is represented by a pair of numbers, known as coordinates. The first number of the pair (), corresponds to a position on the \( x \)-axis, and the second number (), tells you the position on the \( y \)-axis. This system allows for a clear and precise way to display relationships between variables when graphing equations such as lines, curves, and other shapes. Whether plotting a single point or a line like \( y = 3 \), understanding how the coordinate plane works is key to accurately graphing and interpreting results.

plotting points

Plotting points is a crucial step in graphing any line or curve on the coordinate plane. This process involves identifying and marking specific coordinates that correspond to values in your equation. To graph an equation like \( y = 3 \), begin by identifying a few convenient values of \( x \).
For instance:

• When \( x = 0 \), \( y = 3 \)
• When \( x = 2 \), \( y = 3 \)
• When \( x = -3 \), \( y = 3 \)

Each of these values provides a coordinate point that can be plotted on the plane, such as (0,3), (2,3), and (-3,3). By marking these points, you create a visual guide to connecting them into the desired line—in this case, a horizontal line. After plotting enough points, draw a line connecting them, which should be straight in the case of a linear equation. The process of plotting points converts algebraic equations into meaningful visual representations.