Verified Answer :
- Create a Cartesian coordinate plane with an x-range of \([-8, 6]\) and a y-range of \([-6, 4]\).
- Plot points \(P(-2, 0)\), \(Q(-1, -4)\), \(R(5, 0)\), \(K(0, -1)\), and \(S(0, 4)\) individually by moving along the x and y axes.
- Plot points \(J(6, 3)\), \(K(4, -5)\), \(L(0, -3)\), \(M(-8, -6)\), and \(N(-6, -4)\) based on their Cartesian coordinates.
- Label all points clearly to verify their positions relative to the origin \((0,0)\).
- The final representation is a set of 10 points mapped onto the 2D plane: \(\boxed{\text{Points } P, Q, R, K, S, J, K, L, M, N \text{ plotted on a coordinate grid}}\).
Explanation
- Analyzing and locating the coordinates
To represent these points on a Cartesian coordinate plane, we first establish a standard x-y coordinate system. The horizontal axis (x-axis) and the vertical axis (y-axis) intersect at the origin \((0,0)\). We will scale the axes appropriately to accommodate the given range of x-coordinates (from \(-8\) to \(6\)) and y-coordinates (from \(-6\) to \(4\)).
For the first set of points:
- Point \(P(-2, 0)\) is located on the negative side of the x-axis, 2 units to the left of the origin.
- Point \(Q(-1, -4)\) is located in the third quadrant, 1 unit left and 4 units down from the origin.
- Point \(R(5, 0)\) is located on the positive side of the x-axis, 5 units to the right of the origin.
- Point \(K(0, -1)\) is located on the negative side of the y-axis, 1 unit below the origin.
- Point \(S(0, 4)\) is located on the positive side of the y-axis, 4 units above the origin.
For the second set of points:
- Point \(J(6, 3)\) is located in the first quadrant, 6 units right and 3 units up from the origin.
- Point \(K(4, -5)\) is located in the fourth quadrant, 4 units right and 5 units down from the origin.
- Point \(L(0, -3)\) is located on the negative side of the y-axis, 3 units below the origin.
- Point \(M(-8, -6)\) is located in the third quadrant, 8 units left and 6 units down from the origin.
- Point \(N(-6, -4)\) is located in the third quadrant, 6 units left and 4 units down from the origin.
- Representing the points on a diagram
To visualize these points, draw two perpendicular lines crossing at \((0,0)\). Mark the x-axis with increments from \(-8\) to \(6\) and the y-axis with increments from \(-6\) to \(4\). For each point, move along the x-axis by the first coordinate value, then move up or down by the second coordinate value to plot the position. Label each point clearly next to its position. This diagram provides a clear spatial understanding of how these coordinates relate to each other in the Cartesian plane.
Examples
Understanding how to represent points on a coordinate system is essential for fields like architecture and GPS mapping. For example, if you are mapping the locations of different types of trees in a garden relative to a central fountain at the origin \((0,0)\), you can use these coordinates to precisely place them on your garden plan, ensuring that the layout is accurate and easy to follow.