Problem
5. Suppose that the graph of the function $f(x)=2 x^{2}$ is reflected in the $x$-axis, translated 2 units to the left, and then translated 5 units upward. What could the equation of the quadratic function of the resultant graph be? A. $f(x)=-(x+2)^{2}+5$ B. $f(x)=2(-x-2)^{2}+5$ C. $f(x)=-2(x-2)^{2}+5$ D. $f(x)=-2(x+2)^{2}+5$
Solution
Step 1 ：The original function is \(f(x)=2 x^{2}\).
Step 2 ：Reflecting the graph in the x-axis changes the sign of the function, so we get \(f(x)=-2 x^{2}\).
Step 3 ：Translating 2 units to the left means replacing \(x\) with \((x+2)\), so we get \(f(x)=-2 (x+2)^{2}\).
Step 4 ：Translating 5 units upward means adding 5 to the function, so we get \(f(x)=-2 (x+2)^{2}+5\).
Step 5 ：The equation of the quadratic function of the resultant graph is \(\boxed{f(x)=-2 (x+2)^{2}+5}\). This corresponds to option D.
