Problem 5 - Algebra and Trigonometry

Question

Plot the given points in a coordinate plane: \((2,3),(-2,3),(4,5),(4,-5),(-4,5),(-4,-5)\)

Step-by-Step Solution

Answer

Plot each point by moving horizontally (x-axis) and vertically (y-axis) from the origin.

Step 1 - Understand the Coordinate Plane

A coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is defined by a pair of coordinates \(x,y\), where \(x\) denotes the horizontal position and \(y\) denotes the vertical position.

Step 2 - Plot the Point (2, 3)

To plot the point \( (2, 3) \), start at the origin (0,0). Move 2 units to the right along the x-axis since the x-coordinate is positive. Then, move 3 units upwards along the y-axis because the y-coordinate is positive. Mark this point on the plane.

Step 3 - Plot the Point (-2, 3)

For the point \( (-2, 3) \), start at the origin. Move 2 units to the left along the x-axis because the x-coordinate is negative. Then, move 3 units upwards along the y-axis. Mark this point.

Step 4 - Plot the Point (4, 5)

To plot \( (4, 5) \), begin at the origin. Move 4 units to the right along the x-axis, then 5 units up along the y-axis. Place a point there.

Step 5 - Plot the Point (4, -5)

For \( (4, -5) \), start at the origin, move 4 units right along the x-axis, then 5 units down since the y-coordinate is negative. Plot this point.

Step 6 - Plot the Point (-4, 5)

To plot \( (-4, 5) \), move 4 units left from the origin along the x-axis and 5 units up along the y-axis. Place the point on the graph.

Step 7 - Plot the Point (-4, -5)

Begin at the origin, move 4 units left along the x-axis, and then 5 units down along the y-axis for \( (-4, -5) \). Mark this point.

Key Concepts

Plotting Points

Plotting points on a coordinate plane is a fundamental skill in mathematics. Each point is specified by a pair of numbers, known as coordinates, written as \(x, y\). The first number indicates how far to move horizontally, while the second number shows how far to move vertically. You always start from the origin, which is (0, 0). This is your reference point where the x-axis and y-axis intersect.

To plot a point:

• Begin at the origin.
• Move horizontally according to the \(x\)-coordinate.
• Then move vertically according to the \(y\)-coordinate.
• Mark the point where you land.

Understanding these steps helps you place any given point accurately on the graph.

x-axis

The x-axis is the horizontal line on the coordinate plane. It runs left and right through the origin. Think of it as a number line that you may have seen before, extended in both directions.

Here are some key points about the x-axis:

• Positive x-coordinates move to the right from the origin.
• Negative x-coordinates move to the left from the origin.
• Each point's x-coordinate tells you how far to move left or right.

The x-axis is crucial for determining the horizontal position of any point. Remember, you always move along the x-axis first before adjusting vertically.

y-axis

The y-axis is the vertical line on the coordinate plane. It crosses the x-axis at the origin, traveling up and down the plane.

Important aspects of the y-axis include:

• Positive y-coordinates move upward from the origin.
• Negative y-coordinates move downward from the origin.
• Each point's y-coordinate dictates vertical movement.

This axis is vital for finding the vertical position of a point on the plane. Always make your horizontal move along the x-axis first, then vertical along the y-axis.

Origin

The origin is perhaps the most critical point on a coordinate plane. It is located at (0, 0), where the x-axis and y-axis intersect. Think of it as your home base for plotting any point.

The origin is important for several reasons:

• It serves as a reference point when moving along the axes.
• All points are plotted relative to the origin.
• The coordinates of the origin are always (0, 0).

By understanding the origin, you can navigate the coordinate plane more effectively. Start from the origin to apply movements dictated by the point’s coordinates.