Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

0. Review of College Algebra

Basics of Graphing

Trigonometry

0. Review of College Algebra

Basics of Graphing

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Problem 61

Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.center (√2, √2), radius √2

Verified step by step guidance

Identify the center of the circle as $(\sqrt{2}, \sqrt{2})$ and the radius as $\sqrt{2}$.

Recall the center-radius form of a circle's equation: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Substitute the center $(h, k) = (\sqrt{2}, \sqrt{2})$ and the radius $r = \sqrt{2}$ into the equation.

The equation becomes $(x - \sqrt{2})^2 + (y - \sqrt{2})^2 = (\sqrt{2})^2$.

Simplify the equation to $(x - \sqrt{2})^2 + (y - \sqrt{2})^2 = 2$.

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Key Concepts

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

Center-Radius Form of a Circle

The center-radius form of a circle's equation is expressed as (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 both graphing and analysis of the circle's properties.

Graphing a Circle

Graphing a circle involves plotting its center on a coordinate plane and using the radius to determine the points that lie on the circle. From the center, you can move r units in all directions (up, down, left, right) to find key points, which can then be connected to form the circular shape.

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in a plane. It is given by d = √((x₂ - x₁)² + (y₂ - y₁)²). This concept is essential for understanding how far points are from the center of the circle, which helps in verifying if points lie on the circle.