Understanding domain restrictions is crucial for accurately sketching level curves.
The restriction arises because we’re dealing with the square root function, which requires non-negative inputs. For our problem, we must ensure:
- \(25 - x^{2} - y^{2} \geq 0\)
- This determines the range for which \(x\) and \(y\) values the function is defined.
For every \(c\), the relationship transforms the inequality to \(25 - x^{2} - c^{2} \geq 0\). This shows that as \(c\) increases, the allowable values for \(x\) decrease, tightening the domain.
This domain restriction directly affects the span of the graph, limiting \(x\) and \(y\) to a region defined by each \(c\). Understanding these restrictions helps prevent errors and highlights valid points for drawing.