1. **State the problem:** We want to find the transformation that converts the graph of $f(x) = -x^2 + 9$ into the graph of $g(x) = -4x^2 + 9$. 2. **Compare the functions:** Both functions have the form $y = a x^2 + c$ where $a$ affects the vertical stretch/compression and reflection, and $c$ is the vertical shift. 3. **Analyze the coefficients:** - For $f(x)$, the coefficient of $x^2$ is $-1$. - For $g(x)$, the coefficient of $x^2$ is $-4$. 4. **Effect of coefficient change:** Multiplying $x^2$ by 4 inside the function argument corresponds to a horizontal transformation, but here the coefficient multiplies $x^2$ directly, which affects vertical stretch. 5. **Rewrite $g(x)$ in terms of $f(x)$:** $$g(x) = -4x^2 + 9 = 4(-x^2) + 9 = 4(f(x) - 9) + 9 = 4f(x) - 36 + 9 = 4f(x) - 27$$ 6. **Interpretation:** The graph of $g(x)$ is a vertical stretch of $f(x)$ by a factor of 4, then shifted down by 27 units. 7. **Check horizontal transformations:** Horizontal shrink/stretch would affect the input $x$ as $f(bx)$, but here the input $x$ is unchanged. 8. **Check reflections:** Both $f(x)$ and $g(x)$ have negative leading coefficients, so both are reflected across the x-axis already. No new reflection is introduced. **Answer:** The transformation is a vertical stretch by a factor of 4 (not horizontal shrink/stretch or reflection). Since the question asks specifically about horizontal transformations and reflections, none of the options exactly match the vertical stretch. However, the coefficient change from $-1$ to $-4$ corresponds to a vertical stretch, not a horizontal shrink/stretch or reflection. Therefore, **none of the given options (horizontal shrink, horizontal stretch, reflection across y-axis, reflection across x-axis) correctly describe the transformation.** If forced to choose, the closest is **no horizontal transformation or reflection**; the change is a vertical stretch. **Summary:** The graph of $g(x)$ is a vertical stretch of $f(x)$ by a factor of 4, with no horizontal shrink/stretch or reflection.