Question

Make a complete graph of each function. Find the amplitude, period, and phase shift. $$y=\cos (x-1)$$

Short answer

The amplitude of the function is 1, the period is \(2\pi\), and the phase shift is 1 unit to the right.

Step-by-step solution

Identify the Amplitude

The amplitude of a trigonometric function is the coefficient in front of the cosine or sine function. For the function \(y = \cos(x - 1)\), the coefficient in front of the cosine function is 1. Therefore, the amplitude is 1.

Determine the Period

The period of a cosine function is given by \(\frac{2\pi}{k}\), where \(k\) is the coefficient of \(x\) within the cosine function. For the function \(y = \cos(x - 1)\), there is no coefficient other than 1 affecting \(x\), which means \(k = 1\). Therefore, the period is \(2\pi\).

Calculate the Phase Shift

The phase shift is determined by the horizontal translation of the cosine function and is given by the formula \(-\frac{c}{k}\), where \(c\) represents the constant added or subtracted inside the function. For \(y = \cos(x - 1)\), the \(c = -1\). Given that \(k = 1\), the phase shift is 1 unit to the right.

Graphing the Function

To graph \(y = \cos(x - 1)\), start by plotting the key points of the cosine function on the x-axis: - Start at the phase shift, 1 unit to the right of the origin.- Continue plotting points for one period, which is \(2\pi\) units long.- Take note that the cosine graph starts at its maximum value when there is no phase shift, so in this case, it would start 1 unit to the right at the amplitude of 1.- The graph will have its maximum value at the phase shift, intersect the x-axis at the phase shift plus quarter period, reach its minimum value at the phase shift plus half period, intersect the x-axis again at the phase shift plus three quarters of the period, and then return to the maximum at the end of the period. Use the amplitude to determine the maximum and minimum y-values, which are 1 and -1, respectively.

Key concepts

Amplitude of a Trigonometric Function

The amplitude of a trigonometric function, such as a cosine or sine function, represents half the distance between its maximum and minimum values. It's a measure of how far the graph of the function stretches or compresses vertically from its central axis. In our example, the function given is
\(y = \cos(x - 1)\).
The coefficient in front of the cosine function determines the amplitude. With no coefficient or a coefficient of 1, like in this function, the amplitude is 1.
This means the graph will move 1 unit up to reach its peak and 1 unit down to hit its valley from the central axis, which is usually the x-axis in a cosine function graph. But if the coefficient were, say, 2, the function would stretch to twice the typical height, with an amplitude of 2.

Period of a Cosine Function

The period of a cosine function refers to the length of one complete cycle of the wave, from start to finish, before it begins to repeat itself. For a basic cosine graph, this cycle's length is \(2\pi\) radians.
However, if the function's formula has a coefficient multiplied by the variable x, as in \(y = \cos(kx)\), that coefficient will affect the period.
The formula to find the period would then be \(\frac{2\pi}{k}\). In the example \(y = \cos(x - 1)\), the absence of a coefficient other than 1 for x means that the period remains at the standard \(2\pi\) radians. Therefore, the entire wave pattern of this particular cosine function repeats every \(2\pi\) units along the x-axis.

Phase Shift in Trigonometry

Phase shift in trigonometry describes the horizontal movement of a trigonometric graph along the x-axis. If a trigonometric function is altered by adding or subtracting a constant to the variable x, as in \(y = \cos(x - c)\), this causes the graph to shift horizontally.
The direction and magnitude of this shift are indicated by the constant term, with a positive value shifting the graph to the left and a negative value shifting it to the right.
To calculate the phase shift, we use the formula \(-\frac{c}{k}\), where \(c\) is the added constant, and \(k\) is the coefficient of x (if present). In our exercise where \(c = -1\) and \(k = 1\), the function \(y = \cos(x - 1)\) is shifted 1 unit to the right. This rightward shift means that the starting point of the cosine curve, which is typically at (0, 1), is now at (1, 1).

Cosine Function Graph

The graph of a cosine function reveals a wave-like pattern that starts at a maximum point if there's no phase shift. Given the equation \(y = \cos(x - 1)\), the graph will reflect a horizontal shift to the right by 1 unit due to the phase shift, meaning it will no longer intercept the y-axis at its peak.
Key characteristics to remember when graphing a cosine function include:

• The graph's peak is at the amplitude value.
• It crosses the x-axis at a quarter of the period after the maximum, corresponding to the point where the function value is zero.
• The graph reaches its minimum at half the period.
• It will again cross the x-axis at three-quarters of the period before returning to another peak at the full period.

Using these markers, we can sketch the entire graph across one period, ensuring to keep the period at \(2\pi\) units and the amplitude at 1. The graph will oscillate from the maximum to the minimum vertically, while the phase shift adjusts the starting point horizontally.