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| Method of Plotting Points with Given Coordinates | Identifying Coordinates of a Plotted Point | |
Plotting Points in the Cartesian Plane
Method of Plotting Points with Given Coordinates
Plotting a point in the Cartesian plane means locating its exact position based on its given coordinates $(x, y)$. This involves understanding the roles of the x-coordinate (abscissa) and the y-coordinate (ordinate). The abscissa tells us how far the point is from the y-axis, and the ordinate tells us how far it is from the x-axis. Their signs indicate the direction of movement from the origin.
Steps for Plotting a Point P(x, y)
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Start at the Origin: Always begin the plotting process at the origin, denoted by O(0, 0). This is the point where the horizontal (x-axis) and vertical (y-axis) axes intersect.
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Move along the x-axis (Horizontal Movement): Look at the x-coordinate (abscissa), $x$. This value tells you how many units to move horizontally from the origin.
- If $x$ is positive, move $x$ units to the right along the x-axis.
- If $x$ is negative, move $|x|$ units to the left along the x-axis. (Here, $|x|$ represents the absolute value or positive magnitude of $x$).
- If $x$ is zero, there is no horizontal movement; you remain on the y-axis.
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Move parallel to the y-axis (Vertical Movement): From the position reached in Step 2, look at the y-coordinate (ordinate), $y$. This value tells you how many units to move vertically.
- If $y$ is positive, move $y$ units upwards, parallel to the y-axis.
- If $y$ is negative, move $|y|$ units downwards, parallel to the y-axis. (Here, $|y|$ represents the absolute value or positive magnitude of $y$).
- If $y$ is zero, there is no vertical movement; you remain on the x-axis.
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Mark the Point: The final position reached after completing the horizontal and vertical movements described in Step 2 and Step 3 corresponds to the location of the point P(x, y). Mark this specific intersection of the movements clearly with a dot or cross, and label it with its name or coordinates (e.g., P(x, y)).
In essence, to plot P(x, y), you first move horizontally by $x$ units (right if $x>0$, left if $x<0$) and then vertically by $y$ units (up if $y>0$, down if $y<0$).
Example 1. Plot the points A(4, 3), B(-2, 4), C(-3, -2), D(5, -1), E(0, 3), F(-4, 0) on a Cartesian plane.
Answer:
We will follow the steps outlined above for each given point on a Cartesian coordinate system. First, draw the x-axis and y-axis intersecting at the origin (0, 0) and mark the units on both axes.
- A(4, 3): Starting from the origin (0,0), move 4 units to the right along the positive x-axis. From there, move 3 units upwards parallel to the positive y-axis. Mark this position as point A. Point A lies in the first quadrant ($x>0, y>0$).
- B(-2, 4): Starting from the origin (0,0), move 2 units to the left along the negative x-axis. From there, move 4 units upwards parallel to the positive y-axis. Mark this position as point B. Point B lies in the second quadrant ($x<0, y>0$).
- C(-3, -2): Starting from the origin (0,0), move 3 units to the left along the negative x-axis. From there, move 2 units downwards parallel to the negative y-axis. Mark this position as point C. Point C lies in the third quadrant ($x<0, y<0$).
- D(5, -1): Starting from the origin (0,0), move 5 units to the right along the positive x-axis. From there, move 1 unit downwards parallel to the negative y-axis. Mark this position as point D. Point D lies in the fourth quadrant ($x>0, y<0$).
- E(0, 3): Starting from the origin (0,0), the x-coordinate is 0, so there is no movement along the x-axis (stay on the y-axis). The y-coordinate is 3, so move 3 units upwards along the positive y-axis. Mark this position as point E. Point E lies on the positive y-axis ($x=0, y>0$).
- F(-4, 0): Starting from the origin (0,0), the x-coordinate is -4, so move 4 units to the left along the negative x-axis. The y-coordinate is 0, so there is no vertical movement (stay on the x-axis). Mark this position as point F. Point F lies on the negative x-axis ($x<0, y=0$).
Identifying Coordinates of a Plotted Point
To find the coordinates of a point that is already plotted on the Cartesian plane, we essentially reverse the process of plotting. We need to determine the position of the point relative to the x-axis and the y-axis. The coordinates of a point P are given by the ordered pair $(x, y)$, where $x$ is its abscissa and $y$ is its ordinate.
Steps for Identifying Coordinates of Point P
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Determine the x-coordinate (Abscissa): The x-coordinate of a point is its directed distance from the y-axis. To find it from a plotted point P, draw a line segment from P perpendicular to the x-axis. The point where this perpendicular line intersects the x-axis gives the x-coordinate of P. Alternatively, observe the horizontal position of P. If P is to the right of the y-axis, its x-coordinate is positive; if it is to the left, its x-coordinate is negative. The numerical value is the distance from the y-axis.
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Determine the y-coordinate (Ordinate): The y-coordinate of a point is its directed distance from the x-axis. To find it from a plotted point P, draw a line segment from P perpendicular to the y-axis. The point where this perpendicular line intersects the y-axis gives the y-coordinate of P. Alternatively, observe the vertical position of P. If P is above the x-axis, its y-coordinate is positive; if it is below, its y-coordinate is negative. The numerical value is the distance from the x-axis.
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Form the Ordered Pair: Once both the x-coordinate and the y-coordinate have been determined, write them as an ordered pair $(x, y)$, always listing the x-coordinate first, followed by the y-coordinate, enclosed in parentheses. This ordered pair uniquely identifies the position of the point P on the Cartesian plane.
Points lying on the axes have one coordinate equal to zero: points on the x-axis have a y-coordinate of 0 (form $(x, 0)$), and points on the y-axis have an x-coordinate of 0 (form $(0, y)$). The origin is the unique point $(0, 0)$.
Example 1. Identify the coordinates of the points P, Q, R, S plotted on the Cartesian plane below.
Answer:
We apply the steps to each point shown in the figure:
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Point P:
To find the x-coordinate, drop a perpendicular from P to the x-axis. It intersects the x-axis at $3$. Thus, the x-coordinate is $3$.
To find the y-coordinate, drop a perpendicular from P to the y-axis. It intersects the y-axis at $4$. Thus, the y-coordinate is $4$.
The coordinates of P are $\textbf{(3, 4)}$.
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Point Q:
To find the x-coordinate, drop a perpendicular from Q to the x-axis. It intersects the x-axis at $-4$. Thus, the x-coordinate is $-4$.
To find the y-coordinate, drop a perpendicular from Q to the y-axis. It intersects the y-axis at $2$. Thus, the y-coordinate is $2$.
The coordinates of Q are $\textbf{(-4, 2)}$.
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Point R:
Point R lies on the x-axis. This means its distance from the x-axis is $0$, so its y-coordinate is $0$.
To find the x-coordinate, drop a perpendicular from R to the x-axis (which is R itself). The position on the x-axis is $-5$. Thus, the x-coordinate is $-5$.
The coordinates of R are $\textbf{(-5, 0)}$.
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Point S:
To find the x-coordinate, drop a perpendicular from S to the x-axis. It intersects the x-axis at $2$. Thus, the x-coordinate is $2$.
To find the y-coordinate, drop a perpendicular from S to the y-axis. It intersects the y-axis at $-3$. Thus, the y-coordinate is $-3$.
The coordinates of S are $\textbf{(2, -3)}$.