Problem 41 - Prealgebra

Question

Graph each ordered pair on a coordinate system. $$W(0.25,4)$$

Step-by-Step Solution

Answer

Plot (0.25, 4) by locating 0.25 on the x-axis and 4 on the y-axis, drawing lines from these points to intersect.

Step 1: Understand Ordered Pairs

Ordered pairs are written in the form \((x, y)\), where 'x' is the horizontal coordinate and 'y' is the vertical coordinate. For the ordered pair \(W(0.25, 4)\), 0.25 represents the point on the x-axis, and 4 represents the point on the y-axis.

Step 2: Set Up the Coordinate System

To begin, draw a horizontal line for the x-axis and a vertical line for the y-axis that intersect at the origin (0,0). Label these axes accordingly to represent positive and negative directions if necessary. Provide a suitable scale for each axis, ensuring that 0.25 on the x-axis and 4 on the y-axis can be clearly marked.

Step 3: Locate the x-coordinate

Find 0.25 on the x-axis. Since 0.25 is between 0 and 1, ensure that the marking is precise, perhaps halfway between 0 and 0.5, depending on the scale.

Step 4: Locate the y-coordinate

Find the 4 on the y-axis. This will be straightforward if the scale you use includes whole numbers. Mark the point clearly on the y-axis.

Step 5: Plot the Ordered Pair

From 0.25 on the x-axis, draw a vertical line upwards. From 4 on the y-axis, draw a horizontal line to the right. Mark the point where these lines meet as the point \(W(0.25, 4)\). Ensure that the graph is neat and all lines are drawn lightly until locating the exact point to ensure clarity.

Key Concepts

Understanding the Coordinate System

The coordinate system is a two-dimensional plane that helps us locate points using two numbers, called ordered pairs. These numbers are written as \((x, y)\), where 'x' determines the position along the horizontal axis, called the x-axis, and 'y' determines the position along the vertical axis, known as the y-axis. Remember that:

• The point \((0,0)\) is known as the origin, where the x-axis and y-axis intersect.
• Numbers on the x-axis can be positive or negative. Positive numbers are to the right of the origin, while negative numbers are to the left.
• Numbers on the y-axis can also be positive or negative. Positive numbers are above the origin, and negative numbers are below.

The coordinate system makes it easier to visualize and work with points by providing a structured way to graph them. It's important to understand how to navigate this system for effectively plotting points and solving related problems.

Exploring the x-axis and y-axis

In the coordinate system, the x-axis and y-axis are crucial components. They form the backbone of our graphing exercises. Let's explore them further:

• x-axis: This is the horizontal axis. It runs from left to right and is often the first element we consider in an ordered pair.
• y-axis: This vertical axis runs up and down, intersecting the x-axis at the origin.

The axes are usually labeled with increments that make it easy to plot points precisely. For example, marking every unit or every half unit allows you to place points like \(0.25\) between \(0\) and \(0.5\), and \(4\) right on a whole number line on the y-axis. Make sure to clearly label these axes to communicate direction, ensuring smooth navigation and accurate graphing of points.

The Art of Plotting Points

Plotting points on a coordinate system might seem tricky at first, but it can be mastered with a bit of practice and understanding. Here's how you do it:

• Start by locating the x-coordinate on the x-axis. For example, with the point \((0.25, 4)\), find \(0.25\) which is slightly after \(0\) but before \(0.5\).
• Next, find the y-coordinate on the y-axis, which in our example is \(4\). This is a straightforward point along the vertical line.
• From the \(0.25\) mark on the x-axis draw a vertical line upward.
• Similarly, draw a horizontal line from \(4\) on the y-axis until both lines intersect.
• Mark the intersection as point \(W(0.25, 4)\).

To ensure accuracy, it's helpful to draw temporary lines lightly until you verify the point's location. By following these steps, you'll plot points accurately and efficiently, making graph interpretation and representation crystal clear. Practice is key to becoming comfortable with plotting points, so try graphing various points to reinforce your understanding.