Problem 68 - Algebra A Combined Function

Question

Answer each exercise with true or false. Point (3,0) lies on the \(y\) -axis.

Step-by-Step Solution

Answer

False

Step 1: Understand the Coordinates of the Point

A point in a coordinate plane is represented as \((x, y)\). The first value \(x\) denotes how far to the right (positive) or left (negative) from the y-axis the point is, and the second value \(y\) denotes how far up (positive) or down (negative) from the x-axis the point is.

Step 2: Definition of y-axis

The y-axis is the vertical line on a coordinate plane. Points lying on the y-axis have their x-coordinate equal to 0. This means if a point is on the y-axis, it should have the form \((0, y)\).

Step 3: Check the x-coordinate of Point (3,0)

For the point \((3,0)\), the x-coordinate is 3. This indicates that the point is 3 units to the right of the y-axis.

Step 4: Conclusion About Point (3,0)

Since the x-coordinate of the point is not 0, the point \((3,0)\) does not lie on the y-axis. Instead, it lies somewhere along the x-axis.

Key Concepts

x-coordinate

The x-coordinate is a vital element of understanding the position of a point on a coordinate plane. It is the first number in an ordered pair \((x, y)\), denoting how far left or right a point is from the y-axis. The x-coordinate of a point tells us its horizontal position, where:

• If the value is positive, the point is to the right of the y-axis.
• If the value is negative, the point is to the left of the y-axis.
• If the value is zero, the point lies exactly on the y-axis.

In our specific example, we have the point \(3, 0\). Here, the x-coordinate is 3. This means the point is positioned 3 units to the right of the y-axis, clearly indicating its horizontal placement. Recognizing and interpreting the x-coordinate is crucial in pinpointing exactly where on the plane a point is positioned horizontally.

y-coordinate

The y-coordinate in an ordered pair \(x, y\) describes how far up or down a point is from the x-axis on a coordinate plane. This coordinate defines the vertical positioning of a point, revealing whether the point is above, below, or directly on the x-axis.Consider the value of the y-coordinate:

• A positive y-coordinate indicates the point is above the x-axis.
• A negative y-coordinate means the point is below the x-axis.
• A y-coordinate of zero places the point exactly on the x-axis.

In the example point \(3, 0\), the y-coordinate is 0, which means the point lies directly on the x-axis. Understanding the y-coordinate is essential for knowing a point's vertical alignment and ensures accurate graph plotting.

y-axis

The y-axis is a fundamental part of the coordinate plane, serving as the vertical datum line where horizontal and vertical positions are gauged. It is the vertical line that crosses the x-axis at the origin point \(0, 0\), slicing the plane into two halves.Characteristics of the y-axis:

• Point coordinates with an x-coordinate of 0 lie on the y-axis, which means their ordered pair is in the form \(0, y\).
• The y-axis helps in determining the horizontal distance of a point, which is reflected in its x-coordinate.
• It represents all the points that do not move left or right but solely up and down.

When analyzing whether a point like \(3, 0\) lies on the y-axis, we look for an x-coordinate of 0. Since \(3, 0\) has an x-coordinate of 3, it does not lie on the y-axis, but rather on the x-axis. Grasping the role of the y-axis allows us to accurately analyze and interpret the locations of points in the plane.