Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the graph of \(y=200\).

Step-by-Step Solution

Answer

The graph of \(y=200\) is a horizontal line passing through all points with a y-coordinate of 200.

Step 1: Identify the Type of Graph

The equation \(y=200\) is an equation of a horizontal line in a two-dimensional coordinate system.

Step 2: Plot the Graph

On a graph, this would be a straight line parallel to the x-axis passing through the point at which \(y = 200\).

Step 3: Interpret the Graph

So, any point on this line will have a y-coordinate of 200, regardless of the x-coordinate.

Key Concepts

Coordinates in a Two-Dimensional System

Understanding the coordinates in a two-dimensional system is fundamental to graphing. It involves a pair of numbers, usually written as \( (x, y) \) where \( x \) refers to the position along the horizontal axis, and \( y \) indicates the position along the vertical axis.

The origin, marked with the coordinates \( (0, 0) \), is the reference point where these axes intersect. Moving to the right increases the value of \( x \), while moving up increases the value of \( y \). On the contrary, moving left or down will decrease \( x \) or \( y \) respectively. This system allows us to pinpoint exact locations on the plane using these ordered pairs.

Equation of a Horizontal Line

The equation of a horizontal line in algebra is surprisingly simple, it has the form \( y = k \) where \( k \) is a constant. This means that no matter what value \( x \) takes, \( y \) will always be \( k \).

So, when we describe the graph of \( y = 200 \), we are saying that for any \( x \) value we choose, the \( y \) value will remain at 200. This consistency makes horizontal lines easy to identify and graph. They run parallel to the x-axis and cross the y-axis at the point \( (0, k) \)—in our case, \( (0, 200) \)—meaning they are located \( k \) units above or below the x-axis.

Plotting Graphs in Algebra

Plotting graphs in algebra involves transferring equations onto a graph to visualize their meaning. To graph the horizontal line equation \( y = 200 \) from our exercise, we start by marking a dot at any point where \( y \) is 200 on the y-axis. Since it is a horizontal line, the next step is to draw a straight line parallel to the x-axis that passes through this point.

The beauty of plotting graphs is that it provides a visual representation of all possible solutions to the equation. No matter what x-value you select, the y-value for the equation \( y = 200 \) will always be 200, which is why every point on the line fulfills the equation. This also aids in solving problems that involve finding intercepts, slopes, and understanding the relation between variables.