The function \( f(x)=x^{3} \) has been translated 3 to the right and 5 up. Then it undergoes a vertical stretch by a factor of 2 , followed by a reflection over the \( x- \) \( g(x)= \) axis. What is the new function? Show each step - You can get partial credit.

To transform the function \( f(x) = x^3 \) according to the given instructions, we can break it down step-by-step. 1. **Translation 3 to the right:** Translating a function \( f(x) \) to the right by \( a \) units involves replacing \( x \) with \( x - a \). Therefore, translating \( f(x) = x^3 \) 3 units to the right gives: \[ f(x - 3) = (x - 3)^3 \] 2. **Translation 5 up:** To translate the function upward by \( b \) units, we add \( b \) to the function. Thus, we add 5 to our previous result: \[ g(x) = (x - 3)^3 + 5 \] 3. **Vertical stretch by a factor of 2:** A vertical stretch by a factor of \( k \) can be achieved by multiplying the function by \( k \). Therefore, we multiply \( g(x) \) by 2: \[ g(x) = 2 \left( (x - 3)^3 + 5 \right) = 2(x - 3)^3 + 10 \] 4. **Reflection over the x-axis:** Reflecting a function over the \( x \)-axis means multiplying the entire function by -1. Thus, we perform this step: \[ g(x) = -\left( 2(x - 3)^3 + 10 \right) = -2(x - 3)^3 - 10 \] Putting it all together, the final transformed function is: \[ g(x) = -2(x - 3)^3 - 10 \]