Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Plot the points \(A(5,-2), B(-5,-2), C(5,2), D(-5,2)\), \(E(3,0)\), and \(F(0,3)\) on a coordinate plane.

Step-by-Step Solution

Answer

Plot points A, B, C, D, E, and F by using their given coordinates on a coordinate plane.

Step 1: Understand the Coordinate System

A coordinate plane is composed of two axes: the x-axis (horizontal) and the y-axis (vertical). Each point is represented as \(x, y\). The x-value tells how far to move right (positive) or left (negative) from the origin, while the y-value tells how far to move up (positive) or down (negative).

Step 2: Plot Point A

Locate the x-value of 5 on the x-axis and move vertically to the y-value of -2. Point A is placed at (5, -2).

Step 3: Plot Point B

Move to the x-value of -5, then move vertically to the y-value of -2. Point B is located at (-5, -2).

Step 4: Plot Point C

Find the x-value of 5, then move to the y-value of 2. Point C is at (5, 2).

Step 5: Plot Point D

Travel to the x-value of -5 on the x-axis, then go up to the y-value of 2. This is the location of Point D (-5, 2).

Step 6: Plot Point E

Locate x = 3, and from there, move to y = 0, which means you remain on the x-axis. Point E is at (3, 0).

Step 7: Plot Point F

Move to the origin, from x = 0, and then up to y = 3. That's the location for Point F (0, 3).

Step 8: Conclusion: Verify the Points

Ensure all points are plotted correctly by checking and comparing each plotted position to its respective coordinates.

Key Concepts

Plotting Points

Plotting points on a grid is a fundamental skill in coordinate geometry. It combines understanding directions and distances. Every point on a coordinate plane is represented as a pair \((x, y)\). The first value, \(x\), indicates how far to move along the horizontal line. The second value, \(y\), shows movement along the vertical line.
To plot a point:

• Start at the origin, which is the point \((0, 0)\) where both axes meet.
• From the origin, step horizontally to the \((x)\) value. Positive values move right, while negative values move left.
• Next, adjust vertically to the \((y)\) value. Ascend if positive, descend if negative.

After these steps, mark the place where your journey ends on the grid. This mark is your plotted point.

Practicing this process helps improve your accuracy and proficiency in geometry!

Coordinate Plane

The coordinate plane serves as the foundational landscape for graphing points and lines. It's essential to understand this 2-dimensional space to grasp geometry efficiently. A coordinate plane consists of two main parts:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

These axes split the plane into four sections known as quadrants. Each quadrant helps identify which direction the points move. Further, the axes themselves act as boundaries where positive and negative values can stretch endlessly. Their intersection, known as the origin, marks the center of the coordinate plane. It's here that movement to other points begins. Understanding this setup allows students to visualize and solve mathematical problems with greater ease.

X-axis and Y-axis

The x-axis and y-axis form the heart of any coordinate system. These two axes define where and how to place points on the grid. The x-axis is the horizontal line extending from left to right. Here's what to remember about it:

• A positive x-value indicates a move to the right.
• A negative x-value means a move to the left.

The y-axis, on the other hand, is the vertical line running up and down. Key points include:

• A positive y-value pushes you upward along the axis.
• A negative y-value guides you downward.

Grasping how these two axes operate is crucial for correctly positioning points. Each axis works independently but together, allowing accurate navigation in the coordinate plane.

By mastering how to read and move along these axes, you lay the groundwork for more advanced concepts in coordinate geometry.