What is a coordinate plane?
To understand the coordinate plane we need to first talk about number lines. As we know, number lines can be horizontal and vertical. The intersection of these leads to the formation of a coordinate plane. The x-axis is the term for the horizontal axis. The y-axis is the term for the vertical axis. The origin is the point where the x and y-axes intersect.
An ordered pair is the respective x and y coordinate of a particular point in a coordinate plane. The x – coordinate represents the horizontal distance from the origin along the x-axis. The y – coordinate represents the vertical distance from the origin along the y – axis.
For example, (5 , -7)
In the above example, The first number 5 is the x – coordinate and it marks the horizontal distance from the origin along the x – axis. The second number -7 is the y – coordinate and it denotes the vertical distance from the origin along the negative y – axis.
Quadrants: The intersection of the x and y axis separates the coordinate plane into 4 areas. Each of these is known as a quadrant. In a graph, a quadrant is the area enclosed by the x and y axes; there are four quadrants. Quadrant I is in the top right corner, then quadrants II through IV are in a counterclockwise direction.
Quadrant 1: x and y have positive values.
Quadrant 2: x is negative and y is positive.
Quadrant 3: x and y are negative.
Quadrant 4: x is positive and y is negative
For example; If we plot (1, 2), (-1, 2), (1, -2), (-1, -2) in a coordinate plane, then (1, 2) lies in the first quadrant, (-1, 2) lies in the second quadrant, (-1, -2) lies in the third quadrant and (1, -2) lies in the fourth quadrant.
Line graph: Line graphs are straight line graphs. We make line graphs by connecting several points with straight lines.
Reflecting a point in the coordinate plane
- Reflection along the x – axis: Simply multiply the y coordinate of a given point by -1 to reflect it along the x-axis. That is, change the sign of the y coordinate and keep the x coordinate the same to get the coordinates of the point reflected along the x-axis.
For example, Reflecting (-3, 5) along the x – axis.
So, the reflection of (-3, 5) along the x – axis is (-3, -5).
- Reflection along the y-axis: Simply multiply the x coordinate of the point by -1 to reflect it along the y axis. That is, change the sign of the x coordinate and keep the y coordinate the same to get the coordinates of the point reflected along the y-axis.
For example, Reflecting (-3, -3) along the y – axis.
So, that the reflection of (-3, -3) along the y – axis is (3, -3).
- Reflecting a point along both axes: To reflect a point along both axes, we multiply -1 by both the x and y coordinates. That is, we need to change the signs of both x and y coordinates.
Now that we have learnt about reflections along the x and y axes individually, let’s understand the reflection of a point along both axes using an example.
Reflect the point (5, 2) in both the x-axis and y-axis.
Step 1: First, plot the point (5, 2)
Step 2: Reflecting (5, 2) along the x – axis.
Using the same x – coordinate, 5, take the opposite of the y – coordinate. The opposite of 2 is -2. So, the reflection of (5, 2) along the x – axis is (5, -2).
Step 3: Reflecting (5, -2) the y – axis.
Using the same y – coordinate, -2, take the opposite of the x – coordinate. The opposite of 5 is -5. So, the reflection of (5, -2) along the y – axis is (-5, -2).
Finally, the reflection of (5, 2) along the x – axis and y – axis is (-5, -2).
Polygons in the coordinate plane
A polygon is a geometric figure with at least three straight edges and at least three vertices.
Vertices of polygons can be represented or drawn using ordered pairs. Plot and connect the vertices of a polygon in a coordinate plane to form a polygon.
Step 1: The vertices of the polygon are (2, 3), (-1, 2), (-3, 2), and (-2, -3).
Plot and label the points A (2, 3), B (-1, 2), C (-3, 2), and D (-2, -3) on the coordinate plane.
Step 2: Now join the points to form a polygon.
The polygon formed is an irregular quadrilateral.
Finding distances between points in a coordinate plane
When two points on a coordinate plane have the same x- or y-coordinate, the distance between them can be calculated by counting the units between them.
- Two points with the same x-coordinate will run parallel to the y-axis. Hence, calculating the units on the y-axis between the 2 points will tell us the distance.
- Two points with the same y-coordinate will run parallel to the x-axis. Hence, calculating the units on the x-axis between the 2 points will tell us the distance.
We can also use absolute values to find the distance between two points in a coordinate system.
For example, finding the distance between two points in the same quadrant: Let us find the distances between (-5, 3) and (-3 , 3).
Plot the points (-5, 3) and (-3 , 3) on the coordinate plane and then find the difference between |-5| – |-3 = 2. So, the distance between the two points is 2.
For example, finding the distance between points in the different quadrants: To find the distances between (3, 2) and (3, -3), which are in different quadrants, we plot the points (3, 2) and (3, -3) on the coordinate plane and then find the sum of the y-coordinates to get the distance |2| + |-3| = 5. So the distance is 5 units.
Solved Coordinate Planes Examples
- In a city, the market is located at (2, -4) and the swimming pool is located at (2, 3). The school is located at (1, 6), the sports complex is located at (-2, -4) and the dance academy is located at (-6, 6). Graph the locations using the given vertices on the coordinate plane.
Solution: Plot (2, -4), (2, 3), (1, 6), (-2, -4) in a coordinate plane.
The market (2, -4) lies in the fourth quadrant.
The swimming pool (2, 3) lies in the first quadrant.
The school (1, 6) lies in the first quadrant.
The sports complex (-2, -4) lies in the fourth quadrant.
The dance academy (-6, 6) lies in the second quadrant.
- Find the area and perimeter of the polygon by using the given vertices (-5, 3), (6, 3), (6, 6) and (-5, 6)?
Solution:
Step 1:Plot and label the points A(-5, 3), B(6, 3), C(6, 6) and D(-5, 6) on the coordinate plane. Now join the points to form a polygon. The polygon formed is a rectangle.
Step 3: Distance between A and B is |-5 | + |6| = 11, distance between C and D is |-5| + |6| = 11, distance between B and C is |6| – |3| = 3 and distance between A and D is |6| – |3| = 3,
Step 4: Therefore, perimeter = AB + BC + CD + AD
= 11 + 3 +11 + 3
= 28.
Therefore, area = 3 \(\times\) 11 = 33 square units.
- The given table shows the number of vehicles each house owns along with the number of family members. Find out who has the highest number of vehicles in the street.
Solution:
Plot and label the points A(5, 3), B(3, 1), C(4, 2), D(5, 2) and E(1, 0) on the coordinate plane. Draw the line by connecting the points, then observe who has the most number of vehicles from the coordinate plane. A(5, 3), the first house has most vehicles.
There is no end to the numbers we can write as they progress infinitely. Similarly, number lines don’t end as well. Hence, the Cartesian plane extends in all directions indefinitely. We usually draw arrows at the ends of the axes to demonstrate this.
Time series graphs can be used to plot and analyze how various numerical values have changed over time.