1. The problem is to analyze the function $$f(x) = -\frac{1}{4}(x + 2)^2 - 6$$ and describe the transformations applied to the parent function $$y = x^2$$. 2. The parent function is $$y = x^2$$. The transformations include: - Flip over the x-axis (reflection): multiply by -1 - Vertical stretch by a factor of 4 - Shift left by 2 units - Shift down by 6 units 3. The given function can be rewritten to show the vertical stretch explicitly: $$f(x) = -\frac{1}{4}(x + 2)^2 - 6 = -\left(\frac{1}{4}(x + 2)^2\right) - 6$$ 4. To apply a vertical stretch of 4, multiply the function by 4: $$g(x) = 4 \times f(x) = 4 \times \left(-\frac{1}{4}(x + 2)^2 - 6\right) = - (x + 2)^2 - 24$$ 5. The transformations in order: - Start with $$y = x^2$$ - Shift left 2: $$y = (x + 2)^2$$ - Reflect over x-axis: $$y = -(x + 2)^2$$ - Vertical stretch by 4: $$y = -4(x + 2)^2$$ - Shift down 6: $$y = -4(x + 2)^2 - 6$$ 6. The original function $$f(x) = -\frac{1}{4}(x + 2)^2 - 6$$ actually represents a vertical compression by factor 1/4, not a stretch by 4. To get a vertical stretch of 4, the coefficient should be -4, not -1/4. 7. Summary: The function $$f(x) = -\frac{1}{4}(x + 2)^2 - 6$$ is a parabola opening downward (due to the negative sign), shifted left 2 units and down 6 units, with a vertical compression by factor 1/4. Final answer: The function is a downward opening parabola with vertex at $$(-2, -6)$$, vertically compressed by 1/4, shifted left 2 and down 6.