Textbook Question
Without actually graphing, identify the type of graph that each equation has.
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Ch. 6 - Analytic Geometry
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 7, Problem 41
Identify and sketch the graph of each relation.
Verified step by step guidance1
Start with the given equation: \(3x^2 + 6x + 3y^2 - 12y = 12\).
Divide every term by 3 to simplify the equation: \(x^2 + 2x + y^2 - 4y = 4\).
Group the \(x\) terms and \(y\) terms together to prepare for completing the square: \((x^2 + 2x) + (y^2 - 4y) = 4\).
Complete the square for each group:
- For \(x^2 + 2x\), add and subtract \((\frac{2}{2})^2 = 1\).
- For \(y^2 - 4y\), add and subtract \((\frac{-4}{2})^2 = 4\).
Rewrite the equation as:
\((x^2 + 2x + 1 - 1) + (y^2 - 4y + 4 - 4) = 4\).
Rewrite the perfect square trinomials as squared binomials and move constants to the right side:
\((x + 1)^2 - 1 + (y - 2)^2 - 4 = 4\).
Combine constants on the left and add them to the right side:
\((x + 1)^2 + (y - 2)^2 = 4 + 1 + 4\).
This is the equation of a circle centered at \((-1, 2)\) with radius \(\sqrt{9}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Completing the Square
Completing the square is a method used to rewrite quadratic expressions in a form that reveals their geometric properties. By adding and subtracting appropriate constants, you transform expressions like ax² + bx into perfect square trinomials, which helps identify the center and radius of conic sections such as circles or ellipses.
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Solving Quadratic Equations by Completing the Square
Equation of a Circle
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form allows you to identify and graph the circle by determining its center coordinates and radius.
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Graphing Conic Sections
Graphing conic sections involves identifying the type of curve (circle, ellipse, parabola, or hyperbola) from its equation and plotting key features like center, vertices, and axes. Understanding how to manipulate and interpret the equation is essential for accurately sketching the graph.
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3:08Geometries from Conic Sections