Problem 14 - Algebra 1: Concepts and Skills

Question

Plot and label the ordered pairs in a coordinate plane. $$ A(5,2), B(4,3), C(-2,-4) $$

Step-by-Step Solution

Answer

The points A(5,2), B(4,3) and C(-2,-4) are plotted and labelled in the first and third quadrants of the Cartesian plane.

Step 1: Recognize each ordered pair

Determine the x and y coordinates of each pair. The pairs given are: A(5,2), B(4,3) and C(-2,-4). In each ordered pair, the first value is the x-coordinate and the second value is the y-coordinate. Hence, for A(5,2), '5' is the x-coordinate and '2' is the y-coordinate. Similarly, '4' and '3' are the x and y coordinates of point B and for C, '-2' & '-4' are the x and y coordinates respectively.

Step 2: Plotting the pairs

Next, these ordered pairs needs to be plotted on a Cartesian plane. The point A is plotted at 5 units on the x-axis and 2 units on the y-axis in the first quadrant. B is plotted at 4 units from origin to the right on x-axis (positive) and then moving upward by 3 units on the y-axis in the first quadrant. The coordinates for point C are negative, so it will be plotted 2 units to the left on the x-axis and 4 units down on the y-axis; thus, it will be placed in the third quadrant.

Step 3: Labeling the points

After plotting, these points are labelled as 'A', 'B' and 'C' on the graph. This indicates that the point A is at (5, 2), point B is at (4, 3) and point C lies at (-2,-4) on the Cartesian plane.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional plane formed by two number lines intersecting at a right angle. The horizontal line is called the x-axis and the vertical line is called the y-axis. We use these axes to determine the position of points within the plane. Every position on the plane is described by an ordered pair

• The first number in the pair is the x-coordinate, indicating how far left or right the point is.
• The second number is the y-coordinate, showing how far up or down the point is.

This system allows us to precisely locate any point in a flat surface using numerical values.

Plotting Points

Plotting points on a coordinate plane involves placing a dot at the intersection of the x and y values given in an ordered pair. To plot a point, start at the origin, where the x and y axes meet (0,0).

• Move horizontally to the x-coordinate value.
• From there, move vertically to reach the y-coordinate.

After reaching the correct position, mark the spot on the graph and label it if needed. This makes it easy to see the relationship between points on the plane.

Quadrants

The coordinate plane is divided into four regions, called quadrants, based on the signs of the x and y coordinates. These quadrants are:

• First Quadrant: Both x and y coordinates are positive.
• Second Quadrant: x coordinate is negative, y coordinate is positive.
• Third Quadrant: Both x and y coordinates are negative.
• Fourth Quadrant: x coordinate is positive, y coordinate is negative.

Understanding quadrants helps in quickly determining the approximate location where points will be plotted on the plane.

Cartesian Plane

The Cartesian plane, named after the mathematician René Descartes, is the formal name for the coordinate plane. It is a foundational tool in mathematics and allows for geometric and algebraic representations of points and lines. On this plane:

• The x-axis and y-axis intersect at the origin (0,0).
• Ordered pairs are used to represent points.
• Lines and figures can be drawn by connecting points.

It is a crucial concept for graphing mathematical functions and understanding spatial relationships.