https://file.aiquickdraw.com/mathbot/file/1728953802070ic6m0ryh.jpg

The function you're asked to graph is:

\[
g(x) = -2(x - 5)^2 - 2
\]

This is a transformed version of the basic quadratic function \( f(x) = x^2 \). Let's break down the transformations one by one:

1. **Horizontal Shift**: The \( (x - 5) \) means the graph of \( f(x) = x^2 \) is shifted **5 units to the right**.
2. **Vertical Stretch and Reflection**: The factor of \( -2 \) in front of the squared term does two things:
   - The negative sign reflects the graph across the x-axis (flips it upside down).
   - The 2 causes a **vertical stretch** by a factor of 2, making the graph narrower.
3. **Vertical Shift**: The \( -2 \) outside the squared term moves the graph **2 units down**.

### Graphing Steps:
1. Start with the basic parabola \( f(x) = x^2 \).
2. Shift it 5 units to the right, so the vertex moves from \( (0, 0) \) to \( (5, 0) \).
3. Reflect the parabola across the x-axis (make it open downward).
4. Stretch it vertically by a factor of 2, making it narrower than the original \( f(x) = x^2 \).
5. Finally, shift the whole graph down 2 units, placing the vertex at \( (5, -2) \).

### Final Graph:
- **Vertex**: \( (5, -2) \)
- The parabola opens **downward**.
- It is **narrower** than the standard parabola due to the vertical stretch.

Would you like me to generate an image of the graph based on this information? Let me know if you need further clarification or details!

What transformations are needed to graph the function g(x) = -2(x - 5)^2 - 2?

Learn the transformations needed to graph the quadratic function g(x) = -2(x - 5)^2 - 2. This problem covers key concepts such as horizontal shifts, vertical stretches, reflections, and vertical shifts. Suitable for students in Grades 10-12.

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