The graph of the function $f(x) = x^{-2}$ is shown in the figure. Martina was tasked with drawing the graph of the function $$ g(x) = -f(x + 3) + 2 $$ on the same figure.
Martina constructed the graph of $g$ in the following steps:
(1) Using blue, she drew the graph of the function: $$ f_1(x) = f(x) + 2 $$ She obtained this graph by translating the graph of $f$ upward $2$ units along the $y$-axis.
(2) Next, using green, she drew the graph of the function: $$ f_2(x) = f_1(x + 3) = f(x + 3) + 2 $$ She obtained this graph by translating the blue graph of $f_1$ $3$ units to the left along the $x$-axis.
(3) Finally, using red, she drew the requested graph of $$ g(x) = -f_2(x) = -f(x + 3) + 2. $$ She obtained this graph by flipping the green graph of $f_2$ about the $x$-axis.
Did Martina draw the graph correctly? If not, identify the incorrect step.
No. She made a mistake in step (1). She cannot start by translating the graph upward.
No. She made a mistake in step (1). She should have translated the graph of $f$ downward, not upward.
No. She made a mistake in step (2).
No. She made a mistake in step (3).
Yes. She drew the graph correctly.
Instead of constructions the graph of the function $g(x) = -f(x + 3) + 2$, Martina constructed the graph of a different function: $$ f_4(x) = -f(x + 3) - 2 $$ In step (3), she applied symmetry about the $x$-axis, transforming the graph of $f_2(x) = f(x + 3) + 2$ into the graph of $$ f_4(x) = -f_2(x) = -[f(x + 3) + 2] = -f(x + 3) - 2. $$ To obtain the correct graph of $g$, she would have need to take an additional step and translate the graph of $f_4$ $4$ units upward.
A more effective way to construct the graph of $g$ in as follows:
Construct the graph of $f_1(x) = f(x + 3)$ by translating the graph of $f$ $3$ units to the left along the $x$-axis.
Construct the graph of $f_2(x) = -f_1(x) = -f(x + 3)$ by flipping the graph of $f_1$ about the $x$-axis.
Finally, construct the graph of $g(x) = f_2(x) + 2 = -f(x + 3) + 2$ by translating $f_2$ $2$ units upward along the $y$-axis.
The final red graph shown in the figure is the correct graph of $g(x) = -f(x + 3) + 2$.
