Textbook Question
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 9x2 +25y² - 36x + 50y – 164 = 0
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 8, Problem 50
Identify each equation without completing the square.
Verified step by step guidance1
Start by rearranging the given equation to group the terms involving y together: \( y^2 - 4y - 4x = 0 \).
Notice that the equation is quadratic in terms of y. This means it can be expressed in the standard form of a quadratic equation: \( Ay^2 + By + C = 0 \).
Identify the coefficients for the quadratic terms: Here, \( A = 1 \), \( B = -4 \), and \( C = -4x \).
Recognize that this equation represents a parabola because it is a quadratic equation in y. The presence of the \( x \) term indicates that the parabola is oriented in the xy-plane.
To further analyze the equation, consider rewriting it in a form that highlights its structure, such as \( y^2 - 4y = 4x \), which can help in identifying the vertex and axis of symmetry of the parabola.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0, where a, b, and c are constants. In the given equation, y^2 - 4y - 4x = 0, the variable y is squared, indicating it is a quadratic in y. Understanding the structure of quadratic equations is essential for identifying their properties and solutions.
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Introduction to Quadratic Equations
Standard Form of a Quadratic
The standard form of a quadratic equation is expressed as y = ax^2 + bx + c. This form allows for easy identification of the coefficients and the vertex of the parabola represented by the equation. In the context of the given equation, recognizing how to rearrange it into standard form is crucial for analysis and graphing.
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04:34Converting Standard Form to Vertex Form
Graphing Quadratics
Graphing a quadratic equation involves plotting a parabola on the coordinate plane. The shape and position of the parabola are determined by the coefficients of the equation. Understanding how to identify the vertex, axis of symmetry, and intercepts is vital for accurately graphing the equation derived from y^2 - 4x - 4y = 0.
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06:36Solving Quadratic Equations Using The Quadratic Formula
Related Practice
Textbook Question
Graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36
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Textbook Question
Identify each equation without completing the square. 4x2 - 9y2 - 8x - 36y - 68 = 0
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Textbook Question
In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes.
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Textbook Question
Identify each equation without completing the square. y2 - 4x + 2y + 21 = 0
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Textbook Question
In Exercises 51–56, graph each relation. Use the relation's graph to determine its domain and range.
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