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 with the given equation: \(x^{2} + y^{2} - 6y - 7 = 0\).
2. Group the \(y\) terms together and isolate the constant on the other side: \(x^{2} + (y^{2} - 6y) = 7\).
3. Complete the square for the \(y\) terms. Take half of the coefficient of \(y\), which is \(-6\), divide by 2 to get \(-3\), then square it to get \(9\). Add \(9\) to both sides to keep the equation balanced: \(x^{2} + (y^{2} - 6y + 9) = 7 + 9\).
4. Rewrite the perfect square trinomial as a binomial squared: \(x^{2} + (y - 3)^{2} = 16\).
5. Identify the center and radius of the circle from the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). Here, the center is \((0, 3)\) and the radius is \(\sqrt{16}\).

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting a constant. This technique helps transform the equation of a circle into its standard form, making it easier to identify key features like the center and radius.

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. Writing the equation in this form allows direct identification of the circle's center coordinates and radius, which are essential for graphing.

Graphing Circles

Graphing a circle involves plotting its center (h, k) on the coordinate plane and using the radius r to mark points equidistant from the center. Understanding how to interpret the standard form equation enables accurate sketching of the circle's shape and position.