Problem 77 - Algebra A Combined Function

Question

For Exercises 71 through 75, fill in each blank with "0," "positive," or "negative." For Exercises 76 and 77, fill in each blank with "x"or "y." Point \(\quad\) (0, number) Location \(\quad\) __-axis

Step-by-Step Solution

Answer

y

Step 1: Understand the Coordinates

The point given is in the form \((0, \text{number})\). This means the x-coordinate is 0, and the y-coordinate can be any number.

Step 2: Determine Which Axis

Points that have an x-coordinate of zero lie on the y-axis since they have not moved left or right from the origin. Instead, they have moved up or down depending on the y-coordinate.

Step 3: Fill in the Blank

Based on the explanation, the point \((0, \text{number})\) is located on the y-axis. Therefore, the blank should be filled with "y."

Key Concepts

Understanding the x-coordinate

In a coordinate system, each point is identified by two numbers, often called coordinates. The first number is the x-coordinate. It tells you where to place the point along the horizontal line, called the x-axis. If you imagine a map with streets that run left to right, the x-coordinate is like your position on a street.

• A positive x-coordinate means the point is to the right of the origin (0,0).
• A negative x-coordinate means the point is to the left of the origin.

For example, in the point (5,3), 5 is the x-coordinate. It tells you to move 5 units to the right along the x-axis.

When the x-coordinate is 0, it means the point is directly on the y-axis. Thus, the point hasn't moved left or right but sits right in the middle, ready to be moved up or down based on the y-coordinate.

Comprehending the y-coordinate

The y-coordinate is the second number in an ordered pair. It indicates the position along the vertical line called the y-axis. Returning to our map analogy, the y-coordinate helps you understand how far up or down along the vertical path you should go from a starting point.

• A positive y-coordinate means you move up from the x-axis.
• A negative y-coordinate means you move down.

In the point (5,3), 3 is the y-coordinate. It instructs you to move 3 units up from the point on the x-axis defined by x=5.

When the y-coordinate is any number while x-coordinate is 0, your point exists straight up or down from the origin, precisely along the y-axis, demonstrating how independently x and y influence position.

Familiarizing with the axes

Axes are like the guides of the coordinate system. They provide a reference frame for plotting points in two-dimensional space. There are two main axes in this system:

• The x-axis is horizontal.
• The y-axis is vertical.

These axes intersect at the origin, which is point (0,0). This is where both x and y-coordinates are zero. Every point in the system is located relative to this origin.

The axes are vital for understanding where a point lies. For example, a point (0, number) lies on the y-axis because the x-coordinate is zero, indicating no horizontal movement from the origin. Conversely, if the y-coordinate were zero and the x-coordinate non-zero, the point would reside on the x-axis.

The concept of axes underpins the entire idea of a coordinate plane, enabling precise descriptions of where things are located within a defined area.