We are tasked with finding the transformed function $k(x)$ based on the given function $f(x) = x^2$, which undergoes the following transformations:

1. Translated to the right by 3: This means replacing $x$ with $x - 3$.
2. Translated up by 2: This means adding $2$ to the entire function.
3. Stretched by a factor of $-\frac{1}{2}$: This means multiplying the function by $-\frac{1}{2}$.

Step-by-step transformation:

1. Start with $f(x) = x^2$.
2. Translate to the right by 3: Replace $x$ with $x - 3$. $f(x - 3) = (x - 3)^2$
3. Apply the vertical stretch by $-\frac{1}{2}$: Multiply by $-\frac{1}{2}$. $-\frac{1}{2}(x - 3)^2$
4. Translate up by 2: Add $2$ to the function. $k(x) = -\frac{1}{2}(x - 3)^2 + 2$

Final answer:

$k(x) = -\frac{1}{2}(x - 3)^2 + 2$

Let me know if you'd like a detailed explanation of any step!

Related Questions:

1. What does the stretch factor $-\frac{1}{2}$ do to the graph?
2. How does translating to the right affect the graph of a quadratic function?
3. What happens when a function is translated vertically (up or down)?
4. How would the equation change if the stretch factor were positive $\frac{1}{2}$ instead of negative $-\frac{1}{2}$?
5. How do you find the vertex of the transformed function?

Tip:

When dealing with transformations, always apply horizontal changes first (translations or stretches) before vertical ones!