Graph $y=2 \cos \left(\frac{1}{2} x\right)-1$ in the interactive widget.
Note that one moveable point always defines an extremum point in the graph and the other point always defines a neighbouring intersection with the midline.

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∻Solution by Steps∻ ‖ step 1 ⋮ To graph the function $y = 2\cos\left(\frac{1}{2}x\right) - 1$, we need to consider the amplitude, frequency, phase shift, and vertical shift. ‖ ‖ step 2 ⋮ The amplitude is 2, which means the graph oscillates 2 units above and below the midline. ‖ ‖ step 3 ⋮ The frequency is determined by the coefficient $\frac{1}{2}$ of $x$ inside the cosine function, which results in a period of $T = \frac{2\pi}{\frac{1}{2}} = 4\pi$. ‖ ‖ step 4 ⋮ There is no phase shift as there is no horizontal translation inside the cosine function. ‖ ‖ step 5 ⋮ The vertical shift is -1, which means the midline of the graph is at $y = -1$. ‖ ‖ step 6 ⋮ The graph oscillates between $y = 1$ and $y = -3$, and has a period of $4\pi$. ‖ ‖ step 7 ⋮ The green point at $(0, 2)$ indicates the maximum value of the function within one period. ‖ ∻Answer∻ ⚹ The graph of $y = 2\cos\left(\frac{1}{2}x\right) - 1$ oscillates between $y = 1$ and $y = -3$, with a period of $4\pi$, and has a vertical shift down by 1 unit. ⚹ ∻Key Concept∻ ⚹Graphing Trigonometric Functions⚹ ∻Explanation∻ ⚹To graph a trigonometric function like a cosine function, it's important to identify the amplitude, period, phase shift, and vertical shift. These determine the shape and position of the graph on the coordinate plane.⚹◊What is the period of the cosine function represented in the graph $y=2\cos\left(\frac{1}{2}x\right)-1$?⍭ Generate me a similar question◊

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