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Chapter 2: Problem 15

Sketch a set of coordinate axes and then plot the point. $$ (3,-1) $$

Short Answer

Expert verified
First, draw the coordinate axes, forming a right angle at the origin (0, 0). Then, move 3 units to the right from the origin on the x-axis, and 1 unit downwards on the y-axis since the second coordinate is negative. Plot the point where the coordinates intersect and label it as (3, -1). Confirm that the point is correctly plotted 3 units to the right on the x-axis and 1 unit down on the y-axis.

Step by step solution

01

Draw the coordinate axes

Create the x and y-axes on a piece of paper or a graphic program, crossing each other at a right angle at the point (0, 0), known as the origin.
02

Locate the point on the x-axis

Search for the first number in the given point (3, -1), which refers to the x-coordinate. This means, move 3 units to the right from the origin on the x-axis.
03

Locate the point on the y-axis

Now identify the second number in the given point, which refers to the y-coordinate. The point given is (3, -1), which means from the position on the x-axis where we stopped previously, move down 1 unit as it is negative.
04

Plot the point

Mark the point (3, -1) where the x-coordinate and y-coordinate intersect. Label this point as (3, -1) on your coordinate plane.
05

Review the plotted point

Verify that the point (3, -1) is properly plotted on the plane with coordinates 3 units to the right along the x-axis and 1 unit downward along the y-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Plane
The coordinate plane is a fundamental concept in mathematics and geometry. It's essentially a flat, two-dimensional surface where we can plot points and lines. This plane is defined by two axes that intersect at right angles. The point where these axes meet is called the origin, and its coordinates are expressed as (0, 0). On a coordinate plane, each point is identified by an ordered pair of numbers, often referred to as coordinates.

Here are a few important features of the coordinate plane:
  • It is composed of four quadrants, each representing a different combination of positive and negative values along the x and y-axes.
  • The horizontal axis is known as the x-axis, while the vertical axis is the y-axis.
  • Coordinates are written in the form (x, y). The first number indicates the position along the x-axis, and the second number indicates the position along the y-axis.
Understanding the coordinate plane is fundamental to graphing equations, analyzing geometric figures, and solving numerous mathematical problems.
X-Axis and Y-Axis
In the coordinate plane, the x-axis and y-axis are the two perpendicular lines that form the backbone of any plotting system. These axes help us pinpoint the exact location of points on the plane.

Here's what you need to know about the x-axis and y-axis:
  • The x-axis: This is the horizontal line on the plane. It represents all possible values of x, spanning both positive and negative values.
  • The y-axis: This is the vertical line. It represents all possible values of y, again including both positive and negative values.
  • When plotting a point, you always start from the origin. For the x-axis, move horizontally: to the right for positive x-values and to the left for negative x-values.
  • For the y-axis, move vertically from the mark on the x-axis: up for positive y-values and down for negative y-values.
The intersection of these axes, the origin, sets the basis for referencing all other positions on the plane. Grasping these basics is crucial for effectively plotting and reading coordinates.
Plotting Points
Plotting points on the coordinate plane involves finding the correct location for the given coordinates and marking them accurately. This skill is foundational in mathematics. Let's break it down into simple steps, using the point (3, -1) as an example.

Follow these steps when plotting a point:
  • First, identify the x-coordinate, which for our example is 3. Starting from the origin, move 3 units to the right along the x-axis.
  • Next, look at the y-coordinate, which is -1 in our example. From the position at x=3, move down 1 unit since it's a negative number.
  • Mark this spot on the plane, and label it as (3, -1). This label helps to quickly identify the point later.
  • Double-check by counting the units moved along the x and y-axes to ensure accuracy.
By following these steps, you can accurately plot any point on the coordinate plane. Practice with various coordinates will make you more comfortable with this process, enhancing your confidence in handling coordinate geometry.

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