Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. x² + y² = 16

Verified step by step guidance

1. Recognize that the given equation \(x^{2} + y^{2} = 16\) is the standard form of a circle centered at the origin, where the general form is \(\left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}\) with center \((h, k)\) and radius \(r\).
2. Compare the given equation to the general form: since there are no \((x - h)\) or \((y - k)\) terms, the center is at \((0, 0)\).
3. Identify the radius by taking the square root of the constant on the right side: \(r = \sqrt{16}\).
4. To find the domain, consider all possible \(x\)-values for which \(y\) is real. Since \(y^{2} = 16 - x^{2}\), \(y\) is real only when \$16 - x^{2} \geq 0\(, which implies \)-4 \leq x \leq 4$.
5. Similarly, for the range, consider all possible \(y\)-values. Since \(x^{2} = 16 - y^{2}\), \(x\) is real only when \$16 - y^{2} \geq 0\(, which implies \)-4 \leq y \leq 4$.

Key Concepts

Equation 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. In the given equation x² + y² = 16, the center is at the origin (0,0) and the radius is the square root of 16, which is 4.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to mark points in all directions. The circle is the set of all points equidistant from the center. For x² + y² = 16, plot the center at (0,0) and draw a circle passing through points 4 units away on the x- and y-axes.

Domain and Range of a Circle

The domain of a circle is the set of all possible x-values, and the range is the set of all possible y-values. For the circle x² + y² = 16, the domain and range are both [-4, 4], since the radius limits the x and y values to within 4 units from the center.