Understanding Algebraic Descriptions

Oct 13, 2025 by windailys.com 3 views

Introduction

In coordinate geometry, algebraic descriptions are used to map points from one coordinate system to another. These descriptions are essential in various mathematical applications, including graphing, transformations, and geometry. In this article, we will explore how to identify the algebraic description that maps a given point onto another point.

Understanding Algebraic Descriptions

An algebraic description is a mathematical function that takes a point in one coordinate system and maps it to a point in another coordinate system. It is typically represented in the form of (x, y) → (x', y'), where (x, y) is the original point and (x', y') is the mapped point.

The Problem

We are given two points: (-1, 5) and (6, -3). Our task is to find the algebraic description that maps the point (-1, 5) onto the point (6, -3).

Analyzing the Options

Let's analyze the given options:

A) (x, y) → (x+7, y-8) B) (x, y) → (x-7, y-8) C) (x, y) → (x-8, y+3)

To determine the correct algebraic description, we need to apply each option to the point (-1, 5) and see which one maps it onto the point (6, -3).

Applying Option A

Let's apply option A to the point (-1, 5):

(x, y) → (x+7, y-8) (-1, 5) → (-1+7, 5-8) (-1, 5) → (6, -3)

Option A maps the point (-1, 5) onto the point (6, -3).

Applying Option B

Let's apply option B to the point (-1, 5):

(x, y) → (x-7, y-8) (-1, 5) → (-1-7, 5-8) (-1, 5) → (-8, -3)

Option B does not map the point (-1, 5) onto the point (6, -3).

Applying Option C

Let's apply option C to the point (-1, 5):

(x, y) → (x-8, y+3) (-1, 5) → (-1-8, 5+3) (-1, 5) → (-9, 8)

Option C does not map the point (-1, 5) onto the point (6, -3).

Conclusion

Based on our analysis, we can conclude that the algebraic description that maps the point (-1, 5) onto the point (6, -3) is:

(x, y) → (x+7, y-8)

This algebraic description takes the point (-1, 5) and maps it onto the point (6, -3).

Final Answer

Q: What is an algebraic description in coordinate geometry?

A: An algebraic description is a mathematical function that takes a point in one coordinate system and maps it to a point in another coordinate system. It is typically represented in the form of (x, y) → (x', y'), where (x, y) is the original point and (x', y') is the mapped point.

Q: How do I determine the algebraic description that maps a given point onto another point?

A: To determine the algebraic description, you need to analyze the given points and apply the transformation rules to the original point. You can use the following steps:

Identify the original point and the mapped point.
Determine the change in the x-coordinate and the change in the y-coordinate.
Use the changes to create the algebraic description in the form of (x, y) → (x', y').

Q: What are some common algebraic descriptions in coordinate geometry?

A: Some common algebraic descriptions in coordinate geometry include:

Translation: (x, y) → (x + a, y + b), where (a, b) is the translation vector.
Rotation: (x, y) → (x cos θ - y sin θ, x sin θ + y cos θ), where θ is the angle of rotation.
Reflection: (x, y) → (x - 2a, y - 2b), where (a, b) is the point of reflection.
Scaling: (x, y) → (ax, by), where (a, b) are the scaling factors.

Q: How do I apply an algebraic description to a point?

A: To apply an algebraic description to a point, you need to substitute the original point into the algebraic description and simplify the expression. For example, if the algebraic description is (x, y) → (x + 3, y - 2) and the original point is (2, 4), you would substitute x = 2 and y = 4 into the algebraic description to get:

(2, 4) → (2 + 3, 4 - 2) (2, 4) → (5, 2)

Q: What are some real-world applications of algebraic descriptions?

A: Algebraic descriptions have many real-world applications, including:

Computer graphics: Algebraic descriptions are used to create 3D models and animations.
Engineering: Algebraic descriptions are used to design and optimize systems, such as bridges and buildings.
Navigation: Algebraic descriptions are used to calculate distances and directions between points.
Science: Algebraic descriptions are used to model and analyze complex systems, such as population growth and chemical reactions.

Q: How do I determine the inverse of an algebraic description?

A: To determine the inverse of an algebraic description, you need to swap the x and y coordinates and change the signs of the coefficients. For example, if the algebraic description is (x, y) → (x + 3, y - 2), the inverse would be (x, y) → (x - 3, y + 2).

Q: What are some common mistakes to avoid when working with algebraic descriptions?

A: Some common mistakes to avoid when working with algebraic descriptions include:

Not checking the domain and range of the algebraic description.
Not simplifying the algebraic description before applying it to a point.
Not checking the units of the algebraic description.
Not using the correct algebraic description for the problem at hand.

Conclusion

Algebraic descriptions are a powerful tool in coordinate geometry, and understanding how to work with them is essential for solving problems in mathematics and science. By following the steps outlined in this article, you can determine the algebraic description that maps a given point onto another point and apply it to solve real-world problems.