https://file.aiquickdraw.com/mathbot/file/1727691347736u3pzo5d2.jpg

The problem in the image asks to find the equation of a curve that is mirrored (reflected) across the x-axis and y-axis. The original curve is given by:

\[
y = x^2 - 8x - 4
\]

### Steps to solve:

1. **Reflection over the x-axis**: To reflect a curve over the x-axis, we multiply the \( y \)-coordinates by \(-1\). So the equation becomes:
   
   \[
   y = -(x^2 - 8x - 4)
   \]
   Simplifying:
   
   \[
   y = -x^2 + 8x + 4
   \]

2. **Reflection over the y-axis**: To reflect a curve over the y-axis, we replace \( x \) with \(-x\). Applying this to the above equation:

   \[
   y = -(-x)^2 + 8(-x) + 4
   \]
   Simplifying:

   \[
   y = -x^2 - 8x + 4
   \]

Thus, the equation of the curve after reflection over both the x-axis and the y-axis is:

\[
y = -x^2 - 8x + 4
\]

Would you like any further explanation or details? Here are some related questions to explore:

1. What is the general rule for reflecting functions across axes?
2. How does completing the square affect the transformation of quadratic equations?
3. Can a curve be reflected across any line other than the x- and y-axes?
4. What happens when we reflect a curve across both the x- and y-axes in different orders?
5. How can we find the vertex of the original and reflected curves?

**Tip:** To reflect a function across the x-axis, multiply the whole function by \(-1\). For reflection across the y-axis, replace \( x \) with \(-x\).

Sebuah kurva dengan persamaan y = x^2 - 8x - 4 di cerminkan terhadap sumbu x dan dilanjutkan dengan pencerminan terhadap sumbu y. Persamaan bayangan kurva adalah?

Learn how to reflect the quadratic curve y = x^2 - 8x - 4 across both the x-axis and the y-axis. This problem explores algebraic transformations, including reflections and quadratic functions, and is suitable for students in Grades 9-12.

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