Problem 8 - Prealgebra

Question

Graph each of the following ordered pairs. $$(3,-4)$$

Step-by-Step Solution

Answer

Plot the point (3,-4) by moving right 3 units and down 4 units from the origin.

Step 1: Understand Coordinate Plane

The coordinate plane consists of two axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is the origin, denoted as (0,0). Each ordered pair (x, y) indicates a specific location on this plane.

Step 2: Identify Ordered Pair

The given ordered pair is (3, -4). Here, 3 is the x-coordinate indicating the horizontal position, and -4 is the y-coordinate indicating the vertical position.

Step 3: Locate X-Coordinate

Start at the origin (0,0). Move horizontally to the right by 3 units, as the x-coordinate is positive 3. This position is directly above (or below) the point (3,0).

Step 4: Locate Y-Coordinate

From position (3,0), move vertically downward by 4 units because the y-coordinate is -4. This downward movement will locate the point at (3,-4) on the coordinate plane.

Step 5: Mark the Point

Place a dot or point at the location (3,-4) that you have identified on the coordinate plane. This is the graphical representation of the ordered pair (3,-4).

Key Concepts

Coordinate Plane

A coordinate plane is like a map we use to find specific points using two main directions, horizontal and vertical. It consists of two number lines: the x-axis, which runs horizontally, and the y-axis, a vertical line. These lines intersect at a point called the origin, which is like zero on our map, denoted as (0,0). We use this plane to easily find and graph locations using coordinates – a bit like playing a game of battleship where every point has a unique address.

Each position on the coordinate plane is described by an ordered pair, written as (x, y), which tells us exactly where to look. Understanding how these pairs work is vital to graphing. Next, let’s explore how these axes are set up to help us navigate the coordinates.

X-axis and Y-axis

The x-axis and y-axis are the foundational lines in a coordinate plane that help us find any given point. Think of the x-axis as the horizontal road, where moving right means you're increasing x values (positive direction) and moving left means you're decreasing them (negative direction).

The y-axis acts like a vertical ladder. When you go up, it increases y values (positive), and going down decreases them (negative).

• X-axis: Horizontal line where x-coordinates are plotted.
• Y-axis: Vertical line where y-coordinates are plotted.
• Origin: Intersection point at (0,0).

Understanding these axes ensures that you can plot various ordered pairs with ease, giving clear insights into navigating through the graph.

Graphical Representation

Graphing is like painting a picture where each point makes up part of the canvas. A point is marked at the spot its coordinates tell us to go, allowing us to visually interpret mathematical information. The ordered pair (3, -4) tells us that we should locate and then mark a point that lies 3 units to the right and 4 units down from the origin.

This representation makes data more understandable. It's like visual storytelling, turning abstract numbers into something you can see and understand at a glance. Graphical representation provides a way to see relationships and patterns that numbers alone cannot show. It transforms lists of data into intuitive graphics with much less effort.

Locating Points on a Graph

Locating points on a graph is as simple as playing a game of follow-the-leader. You start at the origin, (0,0), and move according to the instructions given by the ordered pair's x and y values.

• Start at the Origin: Begin your journey at (0,0).
• Read the X-Coordinate: Move horizontally to the right (if x is positive) or left (if x is negative).
• Read the Y-Coordinate: From there, move vertically up (if y is positive) or down (if y is negative).
• Mark the Point: Once you reach your destination, place a dot to represent the point.

In our example, the pair (3, -4) directed us to go 3 steps right and 4 steps down. By following this simple navigation, you can graph any point and understand its position relative to the rest of the plane. It gives us the control to map our coordinates accurately, adding to our mathematical toolkit.