Problem 69 - Precalculus: A Concise Course

Question

Use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. $$ y=\cos \left(2 \pi x-\frac{\pi}{2}\right)+1 $$

Step-by-Step Solution

Answer

The graph of \(y=\cos (2 \pi x-\frac{\pi}{2})+1\) is similar to the basic cosine function but shifted \(\frac{\pi}{2}\) units to the right and 1 unit upwards. The period of the function is 1, and two full periods are visible from 0 to 2 on the x-axis. The y-values range from 0 to 2.

Step 1: Identify period and shift

First, identify the period and shift of the function using its cosine component. The coefficient of x inside the cosine function is \(2\pi\), which determines the period of the function. The period is given by \(2\pi / 2\pi = 1\). The shift of the function is determined by the term subtracted from x inside the cosine function. In this case, the function is shifted by \(\frac{\pi}{2}\) to the right.

Step 2: Sketch basic cosine function

A basic cosine function starts at a maximum, decreases to a minimum, and then increases back to the maximum. It's symmetric with respect to the y-axis. Sketch a basic cosine function over one period (which is 1 for this function).

Step 3: Apply transformations

Shift the basic cosine function to the right by \(\frac{\pi}{2}\) and add one to all y-values (translating the function upward by 1). Repeat the pattern to include two full periods.

Step 4: Choose an appropriate viewing window

The x-values should extend from 0 to 2 (two full periods) and y-values should be set between 0 and 2 (since the function is shifted up by 1 from the usual range of -1 to 1).

Key Concepts

Cosine Function Transformations

Trigonometric functions like the cosine function are central to understanding periodic behavior in various mathematical contexts. When it comes to graphing these functions, transformations are crucial.

Imagine a basic cosine graph, which typically starts at its maximum value, descends to a minimum, and then ascends back to the maximum, creating a wave-like pattern. With transformations, we can alter this graph in several ways. The equation given, y = \( \cos(2\pi x - \frac{\pi}{2})\) + 1<\/em>, features two main transformations: a horizontal shift and a vertical translation.

Horizontal shifts occur when we add or subtract a value from the variable x within the cosine function. In the example, there's a shift to the right by \(\frac{\pi}{2}\).

Understanding Horizontal Shifts<\/h4> The horizontal shift is recognized by the subtraction of \(\frac{\pi}{2}\) from the x inside the cosine function, indicating that the graph will start its cycle \(\frac{\pi}{2}\) units to the right of the origin.

Vertical translations are the result of adding or subtracting a value from the entire function. Our function is translated upward by 1 unit.

Comprehending Vertical Translations<\/h4> This transformation moves the entire graph up by one unit, so the new range of the function will be shifted to reflect this change. Instead of oscillating between -1 and 1, the graph now oscillates between 0 and 2.

• A cosine function transformation helps to model real-world behaviors with modified waves.
• Manipulating the period and phase shift can tailor the function to fit specific criteria.

Periodic Functions

Periodic functions are the heartbeats of trigonometry, repeating their values at regular intervals. These functions are essential in fields ranging from engineering to music.

The cosine function, being one of these periodic phenomena, repeats every \(2\pi\) radians. However, when we multiply x by a constant, as seen in \(\cos(2\pi x)\), we effectively alter this repetition interval, known as the period.

The Period of a Transformed Cosine Function<\/h4>For the function y = \(\cos(2\pi x - \frac{\pi}{2})\) + 1, the period is modified by the coefficient of x. Here, the period is \(\frac{2\pi}{2\pi}\), simplifying to 1. This means the waveform repeats itself every unit interval on the x-axis, dictating the regularity with which the pattern appears.

• A thorough understanding of periods can help predict and analyze repetitive patterns in various contexts.
• Adjusting the period is a critical tool in shaping how a trigonometric function behaves over time.

Recognizing the period allows us to predict the function's behavior and determine its graph's wavelength.

Graphing Utilities

With technology being an integral part of education, graphing utilities have become indispensable to students and educators alike. These tools allow for precise and clear visualization of mathematical concepts, such as trigonometric functions.

Graphing utilities range from online applications to sophisticated calculators. They can handle complex equations and provide vital insight into the behavior of functions. When tasked with graphing y = \(\cos(2\pi x - \frac{\pi}{2})\) + 1, these utilities take the guesswork out of the process.

Optimizing the Use of Graphing Tools<\/h4>By inputting the function into a graphing utility, students can easily see the effect of the transformations and how the function behaves over multiple periods. Accurate adjustments to the viewing window, based on the period and shifts required for the function, ensure that all relevant parts of the graph are displayed. In this case, picking an x-range that includes at least two full periods and a y-range that encompasses vertical shifts is essential.

• Using graphing utilities saves time and enhances comprehension of complex patterns.
• They provide interactive opportunities for students to experiment with equations and observe the results.

Function Shifts

Shifts in functions are akin to moving a painting on a wall; the image remains the same, but its position changes. Similarly, mathematics allows us to shift graphs horizontally or vertically without altering their shape.

For the cosine function y = \(\cos(2\pi x - \frac{\pi}{2})\) + 1, we see a clear example of both types of shifts. Moving a function to the right or left is termed a horizontal shift, while moving it up or down is a vertical shift.

Impact of Horizontal and Vertical Shifts<\/h4> The horizontal shift of \(\frac{\pi}{2}\) to the right causes the function's starting point and repeating pattern to commence further along the x-axis. The vertical shift of 1 unit up raises the entire function's graph by that amount. These shifts are crucial for aligning the function with given data points or accommodating specific boundary conditions in various applications.

• Function shifts are vital in tailoring graphs to fit exact specifications or conditions.
• Understanding shifts is fundamental when predicting the behavior of dynamic systems modeled by trigonometric functions.