The function \( f(x)=(x+5)^{3} \) undergoes a shift up of 3 . Then it is reflected over the \( x \)-axis, It then has a vertical stretch by a factor of 4 . Finally, it is shifted \( g(x)= \) right by 3. What is the new function? Show each step. You can get partial credit.

Let's go through each transformation step by step, starting from the function \( f(x) = (x + 5)^3 \). 1. **Shift up by 3**: To shift the function up by 3, we add 3 to the entire function: \[ f(x) + 3 = (x + 5)^3 + 3 \] 2. **Reflect over the \( x \)-axis**: To reflect the function over the \( x \)-axis, we multiply the entire function by -1: \[ -\left((x + 5)^3 + 3\right) = -(x + 5)^3 - 3 \] 3. **Vertical stretch by a factor of 4**: To apply a vertical stretch by a factor of 4, we multiply the entire function by 4: \[ 4\left[-(x + 5)^3 - 3\right] = -4(x + 5)^3 - 12 \] 4. **Shift right by 3**: To shift the function right by 3, we replace \( x \) with \( x - 3 \): \[ g(x) = -4((x - 3) + 5)^3 - 12 = -4(x + 2)^3 - 12 \] Now, combining all these steps, the final function is: \[ g(x) = -4(x + 2)^3 - 12 \]