Problem 41 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=\cot \left(2 x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The period is \(\frac{\pi}{2}\); graph is a shifted cotangent with period \(\frac{\pi}{2}\).

Step 1: Understand the Cotangent Function

The function given is based on the cotangent function, which is defined as \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). The graph repeats every \(\pi\) for \(\cot(x)\) in its simplest form.

Step 2: Determine the Period of the Function

For \(\cot(bx)\), the period is \(\frac{\pi}{b}\). Here \(b = 2\), so the period is \(\frac{\pi}{2}\). Thus, the period of the function \(y = \cot(2x - \frac{\pi}{2})\) is \(\frac{\pi}{2}\).

Step 3: Identify Horizontal Shift

The term \(-\frac{\pi}{2}\) inside the function contributes to a horizontal shift. The expression \(2x - \frac{\pi}{2}\) can be simplified: factor out the 2 to get \(2(x - \frac{\pi}{4})\), indicating a horizontal shift of \(\frac{\pi}{4}\) to the right.

Step 4: Sketch the Graph

To sketch the graph of \(y = \cot(2x - \frac{\pi}{2})\), plot key points and asymptotes. The function has a period of \(\frac{\pi}{2}\) and is horizontally shifted right by \(\frac{\pi}{4}\). This means the first asymptote will be at \(x = \frac{\pi}{4}\). Subsequent vertical asymptotes occur every \(\frac{\pi}{2}\) units.

Step 5: Plot Key Points

Determine points where the function value is zero and infinity. The zeros of \(\cot(x)\) occur where \(x\) is an odd multiple of \(\frac{\pi}{2}\). From \(2x - \frac{\pi}{2} = k\pi\) (where \(k\) is an integer), solve for \(x\) to find specific points, considering the horizontal shift.

Step 6: Final Step: Confirm Your Graph

With asymptotes, zero points, and the period confirmed, draw the function over one period and extend it to show periodicity. The graph should show the typical shape of a cotangent function, shifted and adjusted according to our findings.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is defined as the ratio of the cosine function to the sine function: \[ \cot(x) = \frac{\cos(x)}{\sin(x)} \] This is an important trigonometric function that is the reciprocal of the tangent function. The cotangent function is undefined when \( \sin(x) = 0 \), leading to vertical asymptotes at these points. Generally, these are located at integer multiples of \( \pi \). The graph of the basic cotangent function starts at a vertical asymptote, intersects the x-axis at odd multiples of \( \frac{\pi}{2} \), and ends at the next asymptote, making one complete cycle. Unlike sine and cosine waves, which are continuous, the graph of \( \cot(x) \) consists of distinct branches alternating up and down, repeating every \( \pi \) units. Understanding how the cotangent function behaves is essential before any transformation is applied to it.

Period of a Function

The period of a function effectively describes how often the function's values repeat. For trigonometric functions like the sine, cosine, and cotangent, identifying the period is paramount to understanding their graphs. The period of the cotangent function \( \cot(bx) \) is given by the formula: \[ \text{Period} = \frac{\pi}{b} \] where \( b \) is the coefficient of \( x \) inside the argument of the cotangent. In the example provided, the function \( y = \cot(2x - \frac{\pi}{2}) \) has \( b = 2 \). Plugging this into the formula gives us a period of \( \frac{\pi}{2} \). This indicates that the graph of the function completes one full cycle over an interval of \( \frac{\pi}{2} \), after which it repeats its behavior. This shortened period compared to the standard loop of \( \pi \) in \( \cot(x) \) means that the cycles are more frequent, impacting how the function is sketched.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding both their basic shapes and any modifications due to transformations. For \( y = \cot(2x - \frac{\pi}{2}) \), several steps allow us to accurately plot the graph. - **Identify Transformations:** The period is \( \frac{\pi}{2} \), as calculated earlier. To graph it, notice the shift in the function argument: \( 2x - \frac{\pi}{2} \) can be rewritten as \( 2(x - \frac{\pi}{4}) \). This implies a horizontal shift of \( \frac{\pi}{4} \) to the right. - **Plot Asymptotes:** Vertical asymptotes occur where the cotangent function is undefined. For this transformed function, these asymptotes start at \( x = \frac{\pi}{4} \) and repeat every \( \frac{\pi}{2} \). - **Determine Zero Points:** Zeros occur where the argument yields an odd multiple of \( \frac{\pi}{2} \). Solving for \( x \), given the function's period and shift, will help locate these x-axis intersections. - **Sketching:** Draw the cotangent curve between consecutive vertical asymptotes, showing its characteristic drop from top right to bottom left within each period interval. This full process demonstrates the cyclical nature of cotangent graphs and helps in understanding how each transformation - including compressions, stretches, and shifts - affects the graph.