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

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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 $r$.

Compare the given equation to the general form: since there are no $(x - h)$ or $(y - k)$ terms, the center is at $(0, 0)$.

Identify the radius by taking the square root of the constant on the right side: $r = \sqrt{16}$.

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 $-4 \leq x \leq 4$.

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 $-4 \leq y \leq 4$.

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Key Concepts

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

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.

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