Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 60

Chapter 3, Problem 60

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² - 6y -7=0

Verified step by step guidance

Start with the given equation: \(x^{2} + y^{2} - 6y - 7 = 0\).

Group the \(y\) terms together and isolate the constant on the other side: \(x^{2} + (y^{2} - 6y) = 7\).

Complete the square for the \(y\) terms. Take half of the coefficient of \(y\), which is \(-6\), divide by 2 to get \(-3\), then square it to get \(9\). Add \(9\) to both sides to keep the equation balanced: \(x^{2} + (y^{2} - 6y + 9) = 7 + 9\).

Rewrite the perfect square trinomial as a binomial squared: \(x^{2} + (y - 3)^{2} = 16\).

Identify the center and radius of the circle from the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). Here, the center is \((0, 3)\) and the radius is \(\sqrt{16}\).

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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 the form (x - h)² or (y - k)² by adding and subtracting a constant. This technique helps transform the equation of a circle into its standard form, making it easier to identify key features like the center and radius.

Standard Form 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. Writing the equation in this form allows direct identification of the circle's center coordinates and radius, which are essential for graphing.

Graphing Circles

Graphing a circle involves plotting its center (h, k) on the coordinate plane and using the radius r to mark points equidistant from the center. Understanding how to interpret the standard form equation enables accurate sketching of the circle's shape and position.

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Circles in Standard Form

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