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. $f (x) = - 2 (x + 2)^{2} + 5$

Answer

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Answer

The equation of the quadratic function of the resultant graph is $f (x) = - 2 (x + 2)^{2} + 5$ . This corresponds to option D.

Steps

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 $f (x) = - 2 (x + 2)^{2} + 5$ . This corresponds to option D.