1. **State the problem:** We have a parent function $f(x) = x^2$ and a transformed function $g$ which is translated right 3 units and down 5 units from $f$. We want to find the equation of $g$ in the form $y = ax^2 + bx + c$. 2. **Recall the translation rules:** - Translating right by $h$ units means replacing $x$ by $x - h$ in the function. - Translating down by $k$ units means subtracting $k$ from the function. 3. **Apply the translations:** Since $g$ is $f$ translated right 3 units and down 5 units, we write: $$g(x) = f(x - 3) - 5$$ 4. **Substitute $f(x) = x^2$:** $$g(x) = (x - 3)^2 - 5$$ 5. **Expand the square:** $$g(x) = (x - 3)(x - 3) - 5 = x^2 - 3x - 3x + 9 - 5 = x^2 - 6x + 4$$ 6. **Final equation:** $$g(x) = x^2 - 6x + 4$$ 7. **Match with options:** This matches option B: $y = x^2 - 6x + 4$. **Answer:** B