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 must 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, specifically at \(y = 0\).
- Since the lowest \(y\)-value is \(0\), the range is \(y \geq 0\).
- This does not satisfy the condition \(y \geq -6\) because the range starts at \(0\), not \(-6\).

#### Graph B:
- The graph is a parabola opening downwards.
- The vertex (highest point) of the parabola is below the \(y\)-axis, specifically at \(y = -8\).
- Since the highest \(y\)-value is \(-8\), the range is \(y \leq -8\).
- This does not satisfy the condition \(y \geq -6\) because the range is entirely below \(-6\).

#### Graph C:
- The graph is a parabola opening upwards.
- The vertex (lowest point) of the parabola is at \(y = -6\).
- The graph extends upwards indefinitely, meaning the \(y\)-values start at \(-6\) and go to positive infinity.
- This satisfies the condition \(y \geq -6\) because the range is exactly \(y \geq -6\).

#### Graph D:
- The graph is a parabola opening downwards.
- The vertex (highest point) of the parabola is above the \(y\)-axis, specifically at \(y = 6\).
- Since the highest \(y\)-value is \(6\), the range is \(y \leq 6\).
- This does not satisfy the condition \(y \geq -6\) because the range is limited to values less than or equal to \(6\), and it does not include all values greater than or equal to \(-6\).

Conclusion:

The only graph that satisfies the condition \(y \geq -6\) is Graph C, where the vertex is at \(y = -6\) and the graph extends upwards indefinitely.

Final Answer:

\[
\boxed{C}
\]