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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 53
Chapter 3, Problem 53

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+6x+2y+6 = 0

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Start with the given equation: \(x^{2} + y^{2} + 6x + 2y + 6 = 0\).
Group the \(x\) terms and \(y\) terms together: \((x^{2} + 6x) + (y^{2} + 2y) = -6\) (move the constant to the right side).
Complete the square for the \(x\) terms: take half of 6, which is 3, then square it to get 9. Add 9 inside the \(x\) group.
Complete the square for the \(y\) terms: take half of 2, which is 1, then square it to get 1. Add 1 inside the \(y\) group.
Since you added \(9 + 1 = 10\) to the left side, add 10 to the right side as well to keep the equation balanced. Then rewrite the equation as \((x + 3)^{2} + (y + 1)^{2} = \text{(new right side)}\).

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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 + p)² = q. It involves adding and subtracting a constant to create a perfect square trinomial, which simplifies solving or rewriting equations, especially for conic sections like circles.
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Solving Quadratic Equations by Completing the Square

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. Converting the general form to this form helps identify the circle's key features and makes graphing straightforward.
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Circles in Standard Form

Identifying the Center and Radius

Once the equation is in standard form, the center of the circle is given by the coordinates (h, k), and the radius is the square root of r². These values are essential for graphing the circle accurately and understanding its position and size.
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Identifying Intervals of Unknown Behavior
Related Practice
Textbook Question

Find the average rate of change of f(x) = x^2 - 4x from x_1 = 5 to x_2 = 9.

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Textbook Question

Use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)

Textbook Question

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Textbook Question

In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. f(g(1))

Textbook Question

Find

a. (fog) (x)

b. (go f) (x)

c. (fog) (2)

d. (go f) (2).

f(x) = x+4, g(x) = 2x + 1

Textbook Question

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g (x) = √(x + 2) (x = = −2, −1, 2, 7)