Problem 69 - Algebra A Combined Function

Question

Answer each exercise with true or false. For the point \(\left(-\frac{1}{2}, 1.5\right),\) the first value \(,-\frac{1}{2},\) is the \(x\) -coordinate and the second value, \(1.5,\) is the \(y\) -coordinate.

Step-by-Step Solution

Answer

True

Step 1: Identify the point's components

We have the point \((-\frac{1}{2}, 1.5)\). The first value in the coordinate pair, \(-\frac{1}{2}\), is typically referred to as the \(x\)-coordinate.

Step 2: Examine the second value

The second value in the coordinate pair, \(1.5\), is typically referred to as the \(y\)-coordinate.

Step 3: Confirm the coordinate designation

In the Cartesian coordinate system, the first value is always the \(x\)-coordinate and the second value is the \(y\)-coordinate. Therefore, for the point \((-\frac{1}{2}, 1.5)\), \(-\frac{1}{2}\) is the \(x\)-coordinate and \(1.5\) is the \(y\)-coordinate.

Key Concepts

Understanding the X-Coordinate

In the world of mathematics and geometry, the x-coordinate plays a crucial role when plotting points on a Cartesian plane. When you see a pair of numbers in the form \((x, y)\), the first number is known as the x-coordinate.

The x-coordinate tells you how far along the horizontal axis a point is. It helps to locate a specific position horizontally. Think of it like looking along a ruler that spans left to right. When the x-coordinate has a positive value, the point is to the right of the origin. Conversely, a negative x-coordinate puts the point to the left.

• The x-coordinate is always the first number in the ordered pair.
• It provides the horizontal position relative to the origin (0,0).
• A positive x-coordinate means the point is to the right of the y-axis.
• A negative x-coordinate means the point is to the left of the y-axis.

Understanding the x-coordinate simplifies navigation across the Cartesian plane, letting you visualize the horizontal location of any point.

Exploring the Y-Coordinate

As important as the x-coordinate, the y-coordinate is the second number in an ordered pair \((x, y)\). It helps to determine the vertical position of a point on the Cartesian plane.

The y-coordinate tells you how far up or down the vertical axis a point lies. Imagine a ruler standing vertically that tells you how high or low to look from the origin. If the y-coordinate is positive, the point is above the origin, while a negative y-coordinate places it below.

• The y-coordinate is always the second number in the pair.
• It indicates the vertical distance from the origin (0,0).
• Positive y-values mean the point is above the x-axis.
• Negative y-values place the point below the x-axis.

By knowing the y-coordinate, you can effectively "climb" up or down the Cartesian plane to find your target point's vertical position.

Navigating the Cartesian Plane

The Cartesian plane is a two-dimensional coordinate system where each point is determined by an x-coordinate and a y-coordinate, forming an ordered pair \((x, y)\). Created by René Descartes, this plane is fundamental in mathematics for graphing equations, plotting points, and understanding spatial relationships.

• The horizontal axis is known as the x-axis.
• The vertical axis is the y-axis.
• These two axes intersect at the origin, labeled as \((0,0)\).
• Points are placed using coordinates in the \((x, y)\) format, where x is horizontal and y is vertical.

Each point on this plane has a clear and distinct location due to its unique coordinates. Understanding how to use this plane allows you to visualize mathematical concepts more concretely. It acts as a map, helping show where each point lies depending on its x and y values.