Teaching Domain and Range in Algebra 1 - Maneuvering the Middle - Free Printable

Problem Analysis:

We are tasked with identifying which graph best represents a function whose range is all real numbers greater than or equal to \(-6\). This means the function's output values (y-values) must satisfy the condition:
\[
y \geq -6
\]

Key Concepts:

1. Range: The set of all possible output values (\(y\)-values) of a function.
2. Graph Interpretation: For a graph to have a range of \(y \geq -6\), the lowest point on the graph must be at \(y = -6\) or above, and the graph should extend upwards indefinitely.

Step-by-Step Evaluation of Each Graph:

#### Graph A:
- The graph is a parabola opening upwards.
- The vertex (lowest point) of the parabola is above the \(y\)-axis, meaning the minimum \(y\)-value is greater than 0.
- Since the minimum \(y\)-value is not \(-6\) or less, this graph does not satisfy the condition \(y \geq -6\).

#### Graph B:
- The graph is a parabola opening downwards.
- The vertex (highest point) of the parabola is below the \(y\)-axis, and it appears to be at \(y = -8\).
- The range of this function includes values less than \(-6\), which violates the condition \(y \geq -6\).

#### Graph C:
- The graph is a parabola opening upwards.
- The vertex (lowest point) of the parabola is at \(y = -6\).
- All other points on the graph are above \(y = -6\), meaning the range is exactly \(y \geq -6\).
- This graph satisfies the given condition.

#### Graph D:
- The graph is a parabola opening downwards.
- The vertex (highest point) of the parabola is above the \(y\)-axis, meaning the maximum \(y\)-value is greater than 0.
- Since the graph does not include any \(y\)-values less than or equal to \(-6\), this graph does not satisfy the condition \(y \geq -6\).

Conclusion:

The only graph that satisfies the condition \(y \geq -6\) is Graph C, as its vertex is at \(y = -6\) and all other points are above this value.

Final Answer:

\[
\boxed{C}
\]