Question

Math

Posted 6 months ago

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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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Answer from Sia

Posted 6 months ago

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.

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