Problem 13 - Prealgebra

Question

Graph each of the following ordered pairs. $$\left(1,-\frac{3}{2}\right)$$

Step-by-Step Solution

Answer

Plot point (1,-1.5) on the graph.

Step 1: Understand the Coordinate System

The coordinate system consists of two axes: the x-axis (horizontal) and the y-axis (vertical). Every point on this system is defined by an ordered pair \((x, y)\). The first number, \(x\), indicates the horizontal position, and the second number, \(y\), indicates the vertical position.

Step 2: Identify the Ordered Pair

Our given ordered pair is \((1, -\frac{3}{2})\). The first number, 1, represents the x-coordinate, and the second number, \(-\frac{3}{2}\), represents the y-coordinate.

Step 3: Plot the X-Coordinate

Start at the origin (0,0) on the graph. Move 1 unit to the right along the x-axis because the x-coordinate is 1.

Step 4: Plot the Y-Coordinate

From the point (1,0) on the x-axis, move \(-\frac{3}{2}\) units downwards along the y-axis, since \(-\frac{3}{2}\) is negative. This means moving 1.5 units down.

Step 5: Mark the Point

After moving 1 unit right and \(-\frac{3}{2}\) units down, mark the point \((1, -\frac{3}{2})\) on the graph. This is the location representing the ordered pair.

Key Concepts

Understanding the Coordinate System

The coordinate system is like a map that helps us locate a point in a two-dimensional space. It consists of two main parts:

• The x-axis: This is the horizontal line that runs left to right.
• The y-axis: This is the vertical line that runs up and down.

These two axes intersect at a point called the origin, which is located at (0,0). Whenever we plot points on this graph, we are using an ordered pair of numbers, written as \( (x, y) \), to define a specific location.

The coordinate system lets us use these pairs to precisely pinpoint the position of any object or value on a graph, much like finding a treasure on a map. Each number tells us how far to travel along the axes to reach the desired spot.

Graphing Points with Confidence

Graphing points means placing them accurately on the coordinate plane, using their ordered pairs as guidance. The ordered pair consists of two numbers: the first is the x-coordinate and the second is the y-coordinate.

• To graph a point, start at the origin (0,0).
• Move horizontally to reach the x-coordinate.
• Once you reach the perspective of the x-coordinate on the x-axis, move vertically to reach the y-coordinate.

For instance, given the ordered pair (1, -\(\frac{3}{2}\)), you'll move 1 unit right from the origin along the x-axis, then 1.5 units down because the y-coordinate is negative. This step-by-step movement ensures accuracy in plotting points on the graph, leading to a clear visual representation of data.

Breaking Down X-Axis and Y-Axis

In the world of graphing, the x-axis and y-axis serve as the foundational lines that guide our understanding of positioning and direction.

The x-axis is the baseline; it runs horizontally and divides the plane into top and bottom halves. When plotting points, you adjust along the x-axis based on the x-coordinate: - A positive x-coordinate means moving right from the origin. - A negative x-coordinate means moving left.

The y-axis is the vertical counterpart. It splits the plane into left and right halves. The y-coordinate guides the movement: - A positive y-coordinate means moving up from the horizontal line. - A negative y-coordinate means moving down.

Understanding the x-axis and y-axis as two intersecting lines that form a grid helps in visualizing how each point relates to these axes. It establishes a mental framework where numerical values translate into precise positions on the graph.