Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x-8y+32=0

Verified step by step guidance

1. Rewrite the equation in the form of a circle equation: \((x-h)^2 + (y-k)^2 = r^2\).
2. Group the \(x\) and \(y\) terms: \(x^2 + 4x + y^2 - 8y = -32\).
3. Complete the square for the \(x\) terms: \(x^2 + 4x\).
4. Complete the square for the \(y\) terms: \(y^2 - 8y\).
5. Rewrite the equation using the completed squares and identify the center \((h, k)\) and radius \(r\).

Key Concepts

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. To identify a circle from a general equation, one must rearrange it into this form. This involves completing the square for both x and y terms, which reveals the center and radius directly.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is essential for rewriting the equation of a circle in standard form. By adding and subtracting the necessary constants, one can isolate the squared terms and identify the center and radius of the circle.

Types of Graphs

In algebra, equations can represent different types of graphs, including circles, points, or no graph at all. A circle exists when the equation can be rearranged into the standard form, while a point occurs when the radius is zero. If the equation leads to a contradiction, it indicates that no graph exists, highlighting the importance of analyzing the equation's structure.