Wantity ML of the trinsficmations performed on \( f(x)=x^{2} \) to erecte the greph \[ G(x)=-2(x-6)^{2}+10 \]

To analyze the transformations performed on the function \( f(x) = x^2 \) to obtain the graph of \( G(x) = -2(x-6)^2 + 10 \), we can break down the transformations step by step. 1. **Start with the basic function**: \[ f(x) = x^2 \] 2. **Horizontal Shift**: The term \( (x - 6) \) indicates a horizontal shift. Specifically, it shifts the graph to the right by 6 units. Thus, we have: \[ f_1(x) = (x - 6)^2 \] 3. **Vertical Stretch and Reflection**: The coefficient \(-2\) in front of \( (x - 6)^2 \) indicates two transformations: - A vertical stretch by a factor of 2 (since the absolute value of the coefficient is greater than 1). - A reflection across the x-axis (because the coefficient is negative). Therefore, we apply this transformation: \[ f_2(x) = -2(x - 6)^2 \] 4. **Vertical Shift**: Finally, the \( +10 \) indicates a vertical shift upwards by 10 units. Thus, we have: \[ G(x) = -2(x - 6)^2 + 10 \] ### Summary of Transformations 1. **Horizontal Shift**: Right by 6 units. 2. **Vertical Stretch**: Stretch by a factor of 2. 3. **Reflection**: Reflect across the x-axis. 4. **Vertical Shift**: Upward by 10 units. These transformations take the original function \( f(x) = x^2 \) to the new function \( G(x) = -2(x - 6)^2 + 10 \).