Problem 50 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the equation \(y = 4\) is a straight line parallel to the x-axis, passing through the point (0,4).

Step 1: Identify the equation

Given equation is \(y = 4\), which clearly indicates that y-coordinate is 4 for every value of x. In other words, regardless of x's position, y will always be the same, at 4.

Step 2: Plot the line

On the coordinate plane, mark a point anywhere on y-coordinate 4. Because every point on this line has a y-coordinate of 4, we can choose an arbitrary 'x' value. Now, draw a straight line that passes through this point parallel to the x-axis. The line represents the solution set for the equation \(y = 4\).

Key Concepts

Graph Equation

Graphing an equation on a rectangular coordinate system involves placing points on a plane to visually represent a mathematical relationship. In our case, the equation is simple: \( y = 4 \). This states that no matter the value of \( x \), \( y \) remains fixed at 4. This special type of equation is called a horizontal line equation. You're essentially illustrating that every point on this line shares the same \( y \)-coordinate of 4.

• Understand that the entire equation is already solved because it doesn't depend on \( x \).
• Visual representation: You're turning an algebraic equation into accessible visual information.

This makes it a straightforward equation to graph since it indicates a constant relationship.

y-coordinate

The \( y \)-coordinate determines a point's vertical position on a graph. In the equation \( y = 4 \), this value is consistent across the board. That's because the \( y \)-coordinate tells us how high or low we are on the plane. In every point that satisfies \( y = 4 \), the height or distance from the x-axis is the same.

• For all points on this line, the \( y \)-coordinate remains unaffected by changes in \( x \).
• This simplicity defines the line's horizontal nature on the graph.

Such constancy helps in quickly understanding where the line will lie when visualizing.

x-axis

The \( x \)-axis is the horizontal line on a graph where the \( y \)-coordinate is zero. However, in our example equation \( y = 4 \), the \( x \)-axis plays a different role. It serves purely as a reference point since our line will run parallel to it.

• The line is not dependent on \( x \) and doesn’t intersect with the \( x \)-axis, showing its parallel relationship.
• Any \( x \)-value is acceptable for our graph, illustrating that \( x \) is unrestricted.

This means no adjustments to the \( x \)-axis are needed because the line's placement is wholly based on \( y = 4 \).