In Exercises 53–64, 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 - 6y -7=0

Verified step by step guidance

1. Start by rearranging the equation to group the y terms together: $x^2+(y^2-6y)=7$.
2. To complete the square for the y terms, take the coefficient of y, which is -6, divide it by 2 to get -3, and then square it to get 9.
3. Add and subtract 9 inside the equation to maintain equality: $x^2+(y^2-6y+9-9)=7$.
4. Rewrite the equation by completing the square: $x^2+(y-3{)}^2-9=7$.
5. Add 9 to both sides to write the equation in standard form: $x^2+(y-3{)}^2=16$.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique involves rearranging the equation and adding a constant to both sides to create a binomial squared. It is particularly useful for converting equations into standard form, which simplifies the identification of key features such as the center and radius of a circle.

Standard Form of a Circle

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius. This format allows for easy identification of the circle's center and radius, which are essential for graphing the circle accurately. Understanding this form is crucial for interpreting the geometric properties of the circle.

Graphing Circles

Graphing circles involves plotting points that satisfy the circle's equation on a coordinate plane. The center of the circle is marked at the point (h, k), and the radius r determines how far from the center the circle extends. Knowing how to graph circles helps visualize their properties and relationships with other geometric figures, making it an important skill in algebra.