https://file.aiquickdraw.com/mathbot/file/1727298192194gdj79lni.jpg

Here is the graph of the function \( g(x) = -2\sqrt{9 - x^2} + 6 \), which matches the given equation from the image. The shape of the graph is a downward-facing semi-circle, as expected for a function based on the square root of \( 9 - x^2 \), shifted upward by 6 units and scaled by a factor of -2.

Would you like further details on how this function was derived, or any other clarification?

Here are 5 related questions to expand your understanding:

1. How does changing the coefficient of the square root (e.g., from -2 to a different value) affect the graph?
2. What effect does modifying the constant inside the square root (e.g., changing 9 to another value) have on the graph?
3. How would shifting the graph vertically by adjusting the constant outside the square root change the graph?
4. How does the domain of \( g(x) \) affect its graph, and why is it limited to the range \([-3, 3]\)?
5. How would the graph change if we replaced the square root with a different root (e.g., cube root)?

**Tip**: For square root functions involving expressions like \( 9 - x^2 \), always consider the domain restrictions (here, \( -3 \leq x \leq 3 \)) since the square root of a negative number is undefined in the real number system.

Which of the following functions describes the graph of g(x) = -2√(9 - x^2) + 6?

Learn how to graph the function g(x) = -2√(9 - x^2) + 6. This problem focuses on graph transformations, including vertical shifts, stretching, and the domain of square root functions. Suitable for students in Grades 9-12.

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