Problem 34 - A Graphical Approach to College Algebra

Question

Answer each question. (a) What is the equation of the \(x\) -axis? (b) What is the equation of the \(y\) -axis?

Step-by-Step Solution

Answer

(a) \(y = 0\); (b) \(x = 0\).

Step 1: Understanding the Coordinate Axes

The coordinate axes consist of the \(x\)-axis and the \(y\)-axis, which intersect at the origin \((0,0)\). The \(x\)-axis is horizontal, and the \(y\)-axis is vertical.

Step 2: Equation of the x-axis

The \(x\)-axis is the line where all points have a y-coordinate of 0. Therefore, the equation of the \(x\)-axis is \(y = 0\).

Step 3: Equation of the y-axis

The \(y\)-axis is the line where all points have an x-coordinate of 0. Therefore, the equation of the \(y\)-axis is \(x = 0\).

Key Concepts

x-axis equation

In the coordinate system, the \( x \)-axis is one of the two crucial axes used to locate a point in the plane. Think of it as the horizontal ruler running left and right. The equation of the \( x \)-axis is given by \( y = 0 \). Why? Because every single point on this line has a \( y \)-coordinate of zero.

• For instance, the points \((1,0)\), \((5,0)\), \((-3,0)\) all lie directly on the \( x \)-axis.
• It's this characteristic of having a \( y \)-value of zero that defines the equation.

You won't always be able to visually see every point on the \( x \)-axis, as they stretch infinitely in both directions. However, any point with a \( y \) of zero is a part of this infinite line.
Understanding this makes looking at graphs much easier, as spotting the \( x \)-axis can quickly help you determine certain values on a graph.

y-axis equation

The \( y \)-axis, another vital part of the coordinate plane, serves as the vertical counterpart to the \( x \)-axis. Imagine it as the vertical ruler that runs up and down through the coordinate plane.
Every point on the \( y \)-axis has an \( x \)-coordinate of zero, defining its equation as \( x = 0 \). This means:

• The points \((0,2)\), \((0,-7)\), and \((0,10)\) all lie on the \( y \)-axis.
• These points share a common trait of having zero in their \( x \)-position.

The \( y \)-axis extends infinitely in both upward and downward directions, passing through the origin.
Knowing where the \( y \)-axis sits can greatly assist in interpreting graphs, as it helps to quickly identify the natural starting point or the placement of the vertical line through the origin.

origin in coordinate plane

The origin is a special point in the coordinate plane, denoted as \((0,0)\). It's where the \( x \)-axis and \( y \)-axis intersect. Think of it as the home base for the grid—it marks the central point where measurements along the \( x \) and \( y \) directions start.
In any plane, the origin holds significant importance due to:

• Being the starting point for plotting any other point on the graph.
• Representing zero along both axes, allowing for straightforward calculations and symmetry.

Whenever drawing or interpreting a coordinate system, you'll always find the origin at this intersection, acting as a reference point from which the rest of the plane is oriented.
Remember, understanding where the origin is helps in recognizing and plotting points accurately and keeping track of distances and directions on the coordinate plane.