Problem 50 - College Algebra Essentials

Question

Graph each equation in a rectangular coordinate system. $$y=4$$

Step-by-Step Solution

Answer

The graph of the equation \(y=4\) is a horizontal line that cuts across the y-axis at \(y=4\).

Step 1: Understand the equation

The equation is \(y=4\). It does not involve any x-components hence it is a straight horizontal line which crosses the y-axis at \(y=4\).

Step 2: Draw a Cartesian Plane

Draw a two-dimensional Cartesian Plane. Mark the values on the y-axis, make sure to include the number 4.

Step 3: Plot the Line

On the plane, draw a horizontal line on the point where \(y=4\). The line extends indefinitely in either directions.

Step 4: Final Verification

Make sure the line represents the equation \(y=4\). Any point on this line will have the y-coordinate 4.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a mathematical approach to visually represent algebraic equations. It comprises two number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point known as the origin, denoted by the coordinates (0,0).

Using this system:

• Points are expressed as ordered pairs \(x, y\).
• The system allows us to graphically depict relationships between variables.
• The axes divide the plane into four quadrants.

When graphing, each point on the plane corresponds to one solution of the equation. The rectangular coordinate system provides a straightforward way to visualize how different equations translate to lines, curves, and more.

Horizontal Lines

Horizontal lines in the Cartesian plane are unique because they have a constant y-value across all x-values. For instance, the equation \(y=4\) defines a horizontal line that crosses the y-axis at \(y=4\) without any slope.

Characteristics of horizontal lines:

• They are parallel to the x-axis.
• Every point along the line shares the same y-coordinate.
• The slope is zero, indicating no vertical change.

This makes understanding and plotting these lines intuitive. Simply locate the y-coordinate line where the equation equalizes and draw the line parallel to the x-axis.

Cartesian Plane

The Cartesian plane is a two-dimensional surface where mathematical equations are plotted. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin.

• Quadrants are formed by the intersections, labeled I to IV in a counter-clockwise fashion.
• Positive x-values lie to the right of the origin, while negative x-values are to the left.
• Positive y-values are above the origin, while negative are below.

This setup allows for a precise graphical representation of equations. By plotting points and lines on the Cartesian plane, we gain insights into the behavior of functions and variable interactions.