Which graph best represents a function with a range of all real numbers greater than or equal to -6?

We are asked to determine which graph best represents a function with a range of all real numbers greater than or equal to –6.

Step 1: Understand the range

The range of a function is the set of all possible output values (y-values).
We want a function whose range is:

> $ y \geq -6 $

This means:
- The lowest value the function can take is –6
- All other outputs must be greater than or equal to –6

So, we're looking for a graph where:
- The lowest point (minimum) is at $ y = -6 $
- The graph goes upward from there (no part of the graph goes below $ y = -6 $)

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Step 2: Analyze each graph

Let’s examine each option:

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#### Graph A
- This is a parabola opening upward
- The vertex (lowest point) is at $ (-5, 0) $
- So, the minimum y-value is 0
- Therefore, the range is $ y \geq 0 $
- ✘ Not correct — it does not include values down to –6

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#### Graph B
- This is a parabola opening downward
- The vertex (highest point) is at $ (0, -4) $
- Since it opens downward, the maximum is $ y = -4 $
- The range is $ y \leq -4 $
- But we need $ y \geq -6 $, and this graph only goes down from –4
- Also, it goes below –6, so it includes values less than –6
- ✘ Incorrect — range is wrong direction and includes values less than –6

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#### Graph C
- This is a parabola opening upward
- The vertex (lowest point) is at $ (0, -6) $
- So the minimum y-value is –6
- The graph increases from there
- Therefore, the range is $ y \geq -6 $
- ✔ This matches exactly what we’re looking for!

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#### Graph D
- This is a parabola opening downward
- The vertex (highest point) is at $ (-5, 0) $
- So the maximum is $ y = 0 $
- The range is $ y \leq 0 $
- It goes downward toward negative infinity
- So it includes values less than –6
- ✘ Incorrect — range includes values below –6 and is not bounded below by –6

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✔ Final Answer:

Graph C is the only one with a minimum value of $ y = -6 $ and all other values above it.

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✔ Correct choice: C