1. **State the problem:** We start with the quadratic function $$f(x) = x^2 - 3x + 4$$ and apply three transformations in order: reflection in the x-axis, horizontal shift 2 units right, and vertical shift 3 units up. We want to find the turning point (vertex) of the transformed function. 2. **Find the vertex of the original function:** The vertex of a quadratic $$ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $$a=1$$ and $$b=-3$$, so $$x = -\frac{-3}{2 \times 1} = \frac{3}{2} = 1.5$$. Find $$f(1.5)$$: $$f(1.5) = (1.5)^2 - 3(1.5) + 4 = 2.25 - 4.5 + 4 = 1.75$$. So the original vertex is at $$\left(1.5, 1.75\right)$$. 3. **Apply reflection in the x-axis:** Reflecting over the x-axis changes $$f(x)$$ to $$-f(x)$$. The vertex y-value changes from $$1.75$$ to $$-1.75$$, so the vertex after reflection is at $$\left(1.5, -1.75\right)$$. 4. **Apply horizontal shift 2 units right:** Replace $$x$$ by $$x - 2$$ in the function. The vertex x-coordinate shifts from $$1.5$$ to $$1.5 + 2 = 3.5$$. The y-coordinate remains $$-1.75$$. So the vertex after horizontal shift is $$\left(3.5, -1.75\right)$$. 5. **Apply vertical shift 3 units up:** Add 3 to the y-coordinate. The vertex y-coordinate changes from $$-1.75$$ to $$-1.75 + 3 = 1.25$$. So the final vertex is $$\boxed{\left(3.5, 1.25\right)}$$. 6. **Answer:** The turning point of the transformed function is $$\left(3.5, 1.25\right)$$, which corresponds to option 4. --- **Summary:** - Original vertex: $$\left(1.5, 1.75\right)$$ - After reflection: $$\left(1.5, -1.75\right)$$ - After horizontal shift: $$\left(3.5, -1.75\right)$$ - After vertical shift: $$\left(3.5, 1.25\right)$$