Questions: graph of one complete period of a cosine curve is given. (a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.) (b) Write an equation that represents the curve in the form y= x cos (k(x-b))

Solution Steps

The amplitude of a cosine function is the maximum value of the function. From the graph, the maximum value is $\frac{1}{2}$. Therefore, the amplitude is: $$\text{Amplitude} = \frac{1}{2}$$

The period of a cosine function is the length of one complete cycle. From the graph, one complete cycle occurs from $-\frac{\pi}{3}$ to $\frac{2\pi}{3}$. The length of this interval is: $$\text{Period} = \frac{2\pi}{3} - \left(-\frac{\pi}{3}\right) = \frac{2\pi}{3} + \frac{\pi}{3} = \pi$$

The horizontal shift (phase shift) is the amount by which the graph is shifted horizontally from the standard cosine function $y = \cos(x)$. The standard cosine function starts at its maximum value at $x = 0$. From the graph, the maximum value occurs at $x = -\frac{\pi}{3}$. Therefore, the horizontal shift is: $$\text{Horizontal Shift} = -\frac{\pi}{3}$$

Final Answer