Graph each function over a two-period interval.
y = 1 - 2 cos ((1/2)x)

Verified step by step guidance

1. Identify the given function: \(y = 1 - 2 \cos\left(\frac{1}{2}x\right)\).
2. Determine the period of the cosine function. Recall that the period of \(\cos(bx)\) is given by \(\frac{2\pi}{b}\). Here, \(b = \frac{1}{2}\), so the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).
3. Since the problem asks to graph over a two-period interval, calculate the interval length: \(2 \times 4\pi = 8\pi\). So, the graph should be drawn for \(x\) in the interval \([0, 8\pi]\) (or any other interval of length \(8\pi\)).
4. Find key points within one period to plot: evaluate \(y\) at \(x = 0, 2\pi, 4\pi\) (these correspond to \(0\), \(\pi\), and \(2\pi\) inside the cosine argument after multiplying by \(\frac{1}{2}\)). Calculate \(y\) values at these points to understand the shape.
5. Use the amplitude and vertical shift to sketch the graph: amplitude is \(2\) (from the coefficient of cosine), and the vertical shift is \(+1\). The cosine function oscillates between \(-1\) and \(1\), so \(y\) oscillates between \$1 - 2(1) = -1\( and \)1 - 2(-1) = 3$. Plot these points and connect smoothly to complete two periods.

Key Concepts

Amplitude and Vertical Shift of Trigonometric Functions

The amplitude of a cosine function is the absolute value of the coefficient before the cosine, indicating the maximum deviation from the midline. The vertical shift moves the entire graph up or down by a constant value, determined by the added or subtracted constant outside the cosine term. In y = 1 - 2 cos((1/2)x), the amplitude is 2 and the vertical shift is +1.

Period of a Cosine Function

The period of a cosine function is the length of one complete cycle and is calculated as 2π divided by the coefficient of x inside the cosine. For y = 1 - 2 cos((1/2)x), the coefficient is 1/2, so the period is 2π ÷ (1/2) = 4π. Graphing over two periods means plotting from 0 to 8π.

Graphing Trigonometric Functions

Graphing involves plotting key points such as maxima, minima, and intercepts based on amplitude, period, and phase shifts. For cosine functions, start at the maximum point, then mark quarter-period intervals to identify zeros and minima. Adjust these points according to vertical shifts and amplitude to accurately sketch the graph.