b) The point \( (-9,-2446) \) is on the transformed function \( f(x)=7\left[\frac{1}{2}(x-5)\right]^{3}-45 \) use the transformation statement the find the original point on the parent function. Justify your answer. [T3], [C2]

To find the original point on the parent function from the transformed function \( f(x) = 7\left[\frac{1}{2}(x-5)\right]^{3}-45 \), we need to analyze the transformations applied to the parent function, which is \( f(x) = x^3 \). ### Step 1: Identify the transformations The function \( f(x) = 7\left[\frac{1}{2}(x-5)\right]^{3}-45 \) can be broken down into several transformations: 1. **Horizontal Shift**: The term \( (x-5) \) indicates a horizontal shift to the right by 5 units. 2. **Horizontal Compression**: The factor \( \frac{1}{2} \) indicates a horizontal compression by a factor of 2. 3. **Vertical Stretch**: The factor of 7 indicates a vertical stretch by a factor of 7. 4. **Vertical Shift**: The term \(-45\) indicates a vertical shift downward by 45 units. ### Step 2: Reverse the transformations To find the original point on the parent function \( f(x) = x^3 \), we need to reverse these transformations step by step. 1. **Vertical Shift**: Start with the point \( (-9, -2446) \). To reverse the vertical shift of -45, we add 45 to the y-coordinate: \[ y' = -2446 + 45 = -2401 \] So, the new point is \( (-9, -2401) \). 2. **Vertical Stretch**: Next, we reverse the vertical stretch by dividing the y-coordinate by 7: \[ y'' = \frac{-2401}{7} = -343 \] So, the new point is \( (-9, -343) \). 3. **Horizontal Compression**: To reverse the horizontal compression, we multiply the x-coordinate by 2. However, we first need to account for the horizontal shift of 5. We will first adjust the x-coordinate: \[ x' = -9 + 5 = -4 \] Now, we apply the horizontal compression: \[ x'' = -4 \cdot 2 = -8 \] 4. **Horizontal Shift**: Finally, we need to account for the initial horizontal shift of 5. We subtract 5 from the x-coordinate: \[ x_{\text{original}} = -8 - 5 = -13 \] ### Step 3: Conclusion The original point on the parent function \( f(x) = x^3 \) corresponding to the transformed point \( (-9, -2446) \) is \( (-13, -343) \). ### Justification The transformations were applied in the following order: 1. The point was shifted down by 45. 2. The result was vertically stretched by a factor of 7. 3. The x-coordinate was shifted right by 5 and then compressed horizontally by a factor of 2. Thus, the original point on the parent function is \( (-13, -343) \).