Task: Graph the function: $$f(x)=3\cos\left(\frac x2+\frac{\pi}{4}\right)$$ Margaret sketched the graph of the function $f$ in the following steps (see the figure):
(1) She declared that the parent function of $f$ is the function: $$f_1 (x)=\cos x$$ and sketched its graph (in green).
(2) She then considered the coefficient $\frac12$ at $x$, which affects the period of the function $f$. She sketched the graph (in blue) of the function: $$f_2 (x)=\cos\frac x2$$ (3) Next, Margaret declared that the graph of the function: $$f_3 (x)=\cos\left(\frac x2+\frac{\pi}{4}\right)$$ is obtained by shifting the graph of $f_2$ along the $x$-axis by a value that is determined by the following modification of the equation for $f_3$: $$f_3 (x)=\cos\left[\frac12\left(x+\frac{\pi}{2}\right)\right]$$ Therefore, the graph of $f_3$ is obtained by shifting the graph of $f_2$ by $\frac{\pi}{2}$ in the positive direction along the $x$-axis.
(4) Finally, she considered the coefficient $3$, which affects the range of the function $f$. She multiplied each value of $f_3$ by $3$, stretching the graph of $f_3$ vertically by a factor of $3$, and obtained the resulting graph (in red) of the function $f$.
Margaret made a mistake in her procedure. In which step did Margaret make a mistake?
The mistake is in step (1). The graph of the function $f_1(x)=\cos x$ does not correspond to the graph of $f_1$ in the figure.
The mistake is in step (2). Margaret incorrectly determined the smallest period of the function $f_2$. The correct smallest period of $f_2$ should be half that of the function $f_1$, not twice as large.
The mistake is in step (3). The graph of $f_3$ should be created by shifting the graph of $f_2$ by a value of $\frac{\pi}{2}$, but in the negative direction of the $x$-axis.
The mistake is in step (4). The coefficient $3$ shifts the graph of the function $f_3$ vertically in the direction of the $y$-axis.
The correct graph of the function $f_3$ is obtained by shifting the graph of the function $f_2$ by $\frac{\pi}{2}$ in the negative direction along the $x$-axis.
