Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Intro to Functions & Their Graphs

College Algebra

3. Functions

Intro to Functions & Their Graphs

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1:42 minutes

Problem 35

Textbook Question

In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (-1, 4), r = 2

Verified step by step guidance

Identify the standard form of the equation of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Substitute the given center \((-1, 4)\) into the equation, replacing \(h\) with \(-1\) and \(k\) with \(4\).

Substitute the given radius \(r = 2\) into the equation.

Write the equation as \((x - (-1))^2 + (y - 4)^2 = 2^2\).

Simplify the equation to \((x + 1)^2 + (y - 4)^2 = 4\).

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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, making it fundamental for graphing and analyzing circles in coordinate geometry.

Coordinates of the Center

The center of a circle is represented by the coordinates (h, k). In this case, the center is given as (-1, 4), meaning the circle is located at the point where x = -1 and y = 4 on the Cartesian plane. Understanding how to interpret these coordinates is essential 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 crucial for determining the size of the circle. In this problem, the radius is given as 2, which means that the circle extends 2 units in all directions from its center, affecting the equation's right side in the standard form.