Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 108

Chapter 3, Problem 108

In Exercises 107–108, write the standard form of the equation of the circle with the given center and radius. Center (-2. 4), r = 6

Verified step by step guidance

Recall the standard form of the equation of a circle:

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

, where

(h, k)

is the center of the circle and

r

is the radius.

Substitute the given center

(-2, 4)

into the formula. This means

h = -2

and

k = 4

Substitute the given radius

r = 6

into the formula. Remember to square the radius, so

r^2 = 6^2

Write the equation by replacing

h

k

, and

r^2

in the standard form:

(x - (-2))^2 + (y - 4)^2 = 6^2

Simplify the equation:

(x + 2)^2 + (y - 4)^2 = 36

. This is the standard form of the equation of the circle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Circle's Equation

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format allows for easy identification of the circle's center and radius, facilitating graphing and analysis.

Coordinates of the Center

The center of a circle is represented by the coordinates (h, k). In this case, the center is given as (-2, 4), meaning the circle is centered at the point where x = -2 and y = 4 on the Cartesian plane. Understanding the center's coordinates is crucial for correctly applying them in the standard form equation.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It is denoted by r and is a critical component in the standard form equation. In this problem, the radius is given as 6, which means that the circle extends 6 units from the center in all directions.

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Circles in Standard Form

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