1. The problem asks to identify the order of transformations applied to the function $f(x) = x^2$ to get $f(-x + 4) + 3$. 2. The original function is $f(x) = x^2$. 3. The transformed function is $f(-x + 4) + 3$. 4. Let's analyze the transformations inside the function first: $f(-x + 4)$. 5. Rewrite $-x + 4$ as $-(x - 4)$ to understand the transformations better. 6. The expression $f(-(x - 4))$ means: - First, translate the input $x$ by 4 units to the right (because of $x - 4$ inside the function). - Then reflect over the y-axis (because of the negative sign outside the parentheses). 7. After these inside transformations, the function becomes $f(-(x - 4)) = (-(x - 4))^2 = (x - 4)^2$ because squaring removes the negative sign. 8. Finally, the $+3$ outside the function means translate the entire graph up by 3 units. 9. So the order of transformations applied to $f(x) = x^2$ to get $f(-x + 4) + 3$ is: - First: translate right 4 units (due to $x - 4$ inside the function) - Second: reflect over the y-axis (due to the negative sign) - Third: translate up 3 units (due to $+3$ outside the function) 10. However, the problem asks for the transformations in the order they occurred and gives options: - Type 1: translate left 4 units - Type 2: translate up 3 units - Type 3: reflect over the y-axis 11. Since the function has $-x + 4$, rewriting as $-(x - 4)$ shows a translation right 4 units, but the problem only gives translate left 4 units as an option. 12. To match the problem's options, note that $f(-x + 4) = f(-(x - 4))$ is equivalent to reflecting over the y-axis first (change $x$ to $-x$), then translating left 4 units (because $-x + 4$ inside the function shifts the graph left by 4 units after reflection). 13. Therefore, the correct order of transformations is: - First: reflect over the y-axis (Type 3) - Second: translate left 4 units (Type 1) - Third: translate up 3 units (Type 2) Final answer: First Transformation: 3 Second Transformation: 1 Third Transformation: 2