Problem 45 - Algebra 2

Question

Write \(x^{2}+y^{2}+6 x-2 y-54=0\) in standard form by completing the square. Describe the transformation that can be applied to the graph of \(x^{2}+y^{2}=64\) to obtain the graph of the given equation.

Step-by-Step Solution

Answer

The graph is a circle centered at (-3, 1) with radius 8, obtained by translating \(x^2 + y^2 = 64\) 3 units left and 1 unit up.

Step 1: Original Equation

The given equation is \(x^{2}+y^{2}+6x-2y-54=0\). We need to convert it into standard form by completing the square.

Step 2: Rearrange Terms

Rearrange the equation separating terms involving \(x\) and \(y\): \(x^2 + 6x + y^2 - 2y = 54\).

Step 3: Complete the Square for x

Take the \(x\) terms: \(x^2 + 6x\). The coefficient of \(x\) is 6, half of this is 3, and squaring it gives 9. Add and subtract 9: \((x^2 + 6x + 9) - 9\).

Step 4: Complete the Square for y

Take the \(y\) terms: \(y^2 - 2y\). The coefficient of \(y\) is -2, half of this is -1, and squaring it gives 1. Add and subtract 1: \((y^2 - 2y + 1) - 1\).

Step 5: Rewrite the Equation

Incorporate the completed squares: \((x + 3)^2 - 9 + (y - 1)^2 - 1 = 54\). Simplify to get: \((x + 3)^2 + (y - 1)^2 = 64\).

Step 6: Identify Standard Form

The equation \((x + 3)^2 + (y - 1)^2 = 64\) is in the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), with center \((-3, 1)\) and radius 8.

Step 7: Transformation Description

The graph of \(x^2 + y^2 = 64\) is a circle centered at the origin with radius 8. The transformation to \((x + 3)^2 + (y - 1)^2 = 64\) involves translating the graph 3 units left and 1 unit upwards.

Key Concepts

Standard Form of a Circle

The standard form equation of a circle is expressed as \[(x - h)^2 + (y - k)^2 = r^2\]where

• \( (h, k) \) represents the center of the circle.
• \( r \) is the radius of the circle.

In our specific problem, after completing the square, we found that \[(x + 3)^2 + (y - 1)^2 = 64\]This matches the standard form where:

• The center \[(-h, -k)\] is \((-3, 1)\).
• The radius \( r \) is \( \sqrt{64} = 8 \).

Understanding this form is crucial for identifying both the position and size of any circle described by an equation. Once the equation is rearranged to match this format, it's straightforward to read off these critical components.

Transformation of Graphs

Transforming graphs involves altering their position, shape, or size. In this exercise, we focused on translating a circle. Starting with the equation \(x^2 + y^2 = 64\),which represents a circle centered at the origin with a radius of 8, we transformed it into \((x + 3)^2 + (y - 1)^2 = 64\).This transformation involves a translation:

• 3 units left
• 1 unit up

Translation does not change the size or shape of the graph, only its location on the coordinate plane.
This kind of transformation is essential in understanding how graphs shift and adjust on the plane while maintaining their inherent geometric properties.

Coordinate Geometry

Coordinate geometry, or analytic geometry, connects algebraic equations to geometric shapes. It allows geometric problems to be solved using algebraic techniques.
In the context of this exercise, completing the square turns an algebraic equation of a circle into its geometric form. This enables us to:

• Directly identify the circle's center and radius.
• Understand the spatial relationship of this circle with respect to the coordinate axes.

Successful manipulation of these equations with respect to their geometric representation deepens our understanding of both geometry and coordinate systems.
These insights are valuable for visualizing complex problems and predicting geometric outcomes.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions or equations to reveal useful information or solve problems.
In this case, we used a technique called "completing the square." It begins by grouping terms, then carefully adding and subtracting strategic constants to form perfect square trinomials.

• For terms in \( x \), the half of the coefficient is used: \( x^2 + 6x \to (x + 3)^2 \).
• For terms in \( y \), the same approach: \( y^2 - 2y \to (y - 1)^2 \).

This method transforms a cumbersome equation into an easily interpretable standard form.
Mastering algebraic manipulation is key to success in solving complex mathematical equations and in transforming equations for clearer understanding or visualization within coordinate geometry.