1at transformations change the graph of \( f(x) \) to the graph of \( g(x) \) ? \( x)=-5 x^{2} g(x)=-20 x^{2}-8 \) The graph of \( g(x) \) is the graph of \( f(x) \) stretched vertically by a factor of \( \frac{1}{4} \) and translated down 8 units. b the graph of \( g(x) \) is the graph of \( f(x) \) stretched vertically by a factor of 4 and translated down 8 units. The graph of \( g(x) \) is the graph \( f(x) \) of stretched vertically by a factor of \( \frac{1}{4} \) and translated up 8 units.

To analyze the transformations from \( f(x) = -5x^2 \) to \( g(x) = -20x^2 - 8 \), we can break it down step by step. 1. **Identify the original function**: \[ f(x) = -5x^2 \] 2. **Identify the transformed function**: \[ g(x) = -20x^2 - 8 \] 3. **Compare the coefficients**: - The coefficient of \( x^2 \) in \( f(x) \) is \(-5\). - The coefficient of \( x^2 \) in \( g(x) \) is \(-20\). To find the vertical stretch factor, we can compare these coefficients: \[ \text{Vertical stretch factor} = \frac{-20}{-5} = 4 \] This means that the graph of \( f(x) \) is stretched vertically by a factor of 4. 4. **Identify the vertical translation**: - The term \(-8\) in \( g(x) \) indicates a downward translation. Since \( g(x) \) has \(-8\) added to it, this means the graph is translated down by 8 units. 5. **Conclusion**: - The graph of \( g(x) \) is the graph of \( f(x) \) stretched vertically by a factor of 4 and translated down 8 units. Thus, the correct transformation is: **b**: The graph of \( g(x) \) is the graph of \( f(x) \) stretched vertically by a factor of 4 and translated down 8 units.