Ch. 2 - Functions and Graphs

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Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problema 111

Capitolo 3, Problema 111

In Exercises 109–111, give the center and radius of each circle. x^2 + y^2 - 4x + 2y - 4 = 0

Guida verificata passo dopo passo

Rewrite the given equation of the circle:

x^2 + y^2 - 4x + 2y - 4 = 0

. Group the terms involving

x

and

y

together:

(x^2 - 4x) + (y^2 + 2y) = 4

Complete the square for the

x

-terms. Take half the coefficient of

x

(which is

-4

), square it, and add it inside the parentheses:

(x^2 - 4x + 4)

. To maintain equality, add

4

to the right-hand side of the equation.

Complete the square for the

y

-terms. Take half the coefficient of

y

(which is

2

), square it, and add it inside the parentheses:

(y^2 + 2y + 1)

. To maintain equality, add

1

to the right-hand side of the equation.

Rewrite the equation with the completed squares:

(x - 2)^2 + (y + 1)^2 = 9

. This is now in the standard form of a circle:

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center and

r

is the radius.

Identify the center and radius from the equation. The center is

(h, k) = (2, -1)

, and the radius is

r = \(\sqrt{9}\) = 3

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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. To identify the center and radius from a general equation, it is often necessary to rearrange the equation into this standard form through completing the square.

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Circles in Standard Form

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique involves manipulating the equation to isolate the variable terms and create a squared term, which simplifies the process of identifying the center and radius of a circle.

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Solving Quadratic Equations by Completing the Square

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0. In the context of circles, the terms involving x and y can be rearranged to form a quadratic equation, which is essential for identifying the geometric properties of the circle, such as its center and radius.

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Introduction to Quadratic Equations

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