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Chapter 2: Problem 14

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

Short Answer

Expert verified
Draw a horizontal x-axis and a vertical y-axis, intersecting at the origin (0,0). Mark the units, 1 unit for each integer value. Move 1 unit to the right from the origin along the x-axis and 3 units up along the y-axis. Plot the point, mark it with a dot, and label it (1,3). Verify the position by ensuring the point is horizontally aligned with (1,0) and vertically aligned with (0,3).

Step by step solution

01

1. Prepare the coordinate axes

First, draw a horizontal line, called the x-axis, and a vertical line, called the y-axis, intersecting at a point called the origin (0,0). These two lines will represent our Cartesian plane. Make sure to label the x-axis and y-axis.
02

2. Mark the units on the axes

Choose an appropriate scale, and mark the units along the x-axis (horizontal) and y-axis (vertical). In this case, the scale can be set 1 unit for each integer value as we are given the point (1,3). These axes will represent the position of the point in the Cartesian plane.
03

3. Locate the point on the axes

The point in our exercise is given as (1,3). On a Cartesian plane, the first value of a point (x-coordinate) represents the horizontal position on the x-axis and the second value of a point (y-coordinate) represents the vertical position on the y-axis. Since the x-coordinate of our point is 1, move 1 unit to the right from the origin along the x-axis. Similarly, since the y-coordinate of our point is 3, move 3 units up from the origin along the y-axis.
04

4. Plot the point

Now that we have moved 1 unit right on the x-axis and 3 units up on the y-axis, plot the point at this position by marking it with a dot. After plotting the point, label it with the given coordinates (1,3) to avoid confusion.
05

5. Verify the position

Finally, verify if the plotted point is in the correct position in the Cartesian plane by checking its x and y coordinates. The point should be horizontally aligned with point (1,0) and vertically aligned with point (0,3). If the plotted point is aligned with both of these points, the point (1,3) has been accurately represented on the Cartesian plane.

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Key Concepts

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

Plotting Points
When plotting points on a graph, it's vital to understand how to interpret the coordinates of a point. A point in a Cartesian plane is expressed as an ordered pair \((x,y)\). This tells you exactly where that point is located with two pieces of information:
  • The first number (x-coordinate) tells you how far to move horizontally from the origin (where the x-axis and y-axis intersect).
  • The second number (y-coordinate) tells you how far to move vertically from the origin.
After figuring out these movements, you mark the point by placing a dot on the graph. Once you've placed the point, label it with its coordinates to keep things organized. Double-checking the position is a fantastic habit, ensuring your point aligns with the specified x and y coordinates.
Coordinate Axes
The foundation of the Cartesian coordinate system is built on the coordinate axes. These comprise two lines on a plane:
  • The x-axis, which runs horizontally, and the y-axis, which runs vertically.
They intersect at the origin point, denoted as (0,0). Each axis is like a number line, with both positive and negative numbers extending in opposite directions from the origin.
Adding labels to these axes is crucial so anyone reading the graph can know what direction is positive or negative. It also helps determine the scale, which is the distance each unit represents on the graph. For simplicity, matching the scale to the values you'll be plotting often simplifies the process.
Graphing in Mathematics
Graphing is a visual way to represent mathematical information and understand relationships between variables. It combines plotting points and interpreting coordinate axes. Every graph starts with understanding these components:
  • Each graph tells a unique story based on the data it represents.
  • Different graphs can be created depending on whether you鈥檙e working with lines, curves, or even complex shapes.
Using graphs makes it easier to see trends, compare different datasets visually, and even solve equations. Begin by setting up the axes and ensure they are labeled and scaled appropriately. Next, place each point methodically based on your calculations. Once all the points are plotted, you can draw lines or curves through them as needed to complete the graph. This technique aids in visualizing mathematical principles and can even make complicated data more accessible.

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