Question

\(y=\cos \frac{\pi x}{4}\)

Short answer

The function \(y=cos\left(\frac{\pi x}{4}\right)\) will graph as waves in the x-y plane, with a period of 8 units, and with maximum points at (n*8, 1), minimum points at (n*8+4, -1), and zeros at (n*8 + 2, 0), where n is any integer.

Step-by-step solution

Understand the function

We are given the equation \(y=\cos \frac{\pi x}{4}\). The function \(\cos(x)\) is a periodic function, with a period of \(2\pi\). This means that it repeats itself every \(2\pi\) units. The period of the cosine function is being divided by \(\frac{\pi}{4}\), which means that the period is going to be multiplied by 4.

Adjust the period

According to the basic definition, one period of the standard cosine function is \(2\pi\). But in our case, the period is changed because the \(x\) is being multiplied by a factor of \(\frac{\pi}{4}\). This will cause the period of the cosine function to be \(2\pi \cdot \frac{4}{\pi} = 8\) rather than \(2\pi\).

Identify the key points of the wave

Now, keeping in mind the new period of our cosine function, we can determine the key points of one wave. The key points normally are the maximum point, the minimum point, and the zeros.

Graph the function

Now, we can graph \(y=\cos\left(\frac{\pi x}{4}\right)\) on the x-y plane. The function will have a wave with maximum points at (n*8, 1) and minimum points at (n*8+4, -1), where n is any integer. There will also be zeros at (n*8 + 2, 0). The wavelength of this pattern will be 8, as calculated before, with peaks and troughs repeating every 8 units.

Key concepts

Periodic Functions

Periodic functions are those functions that repeat their values at regular intervals or periods. The most common examples of periodic functions are trigonometric functions, such as sine and cosine. These functions oscillate between two values over a specific interval, called the period.

For example, the standard cosine function, \(\cos(x)\), has a period of \((2\pi)\), meaning it takes \((2\pi)\) units for the function to complete one full cycle and return to its start value. In our given function \(y=\cos\left(\frac{\pi x}{4}\right)\), the period of the function changes due to the factor of \(\frac{\pi}{4}\). The period becomes 8, indicating that the oscillation is spread over 8 units in the x-direction.

• Understanding periods is crucial for graphing and predicting the behavior of periodic functions.
• Period changes affect the spacing of key points, like peaks and troughs, on the graph.

Function Graphing

Function graphing involves plotting the values of a function on a coordinate system. With trigonometric functions, we often encounter sinusoidal waves, which include sine and cosine. These waves display regular patterns that are essential in various areas of science and engineering.

When graphing \(y=\cos\left(\frac{\pi x}{4}\right)\), it is important to note the adjustments in period and key points:

• The function displays maximum values, known as peaks, at points where \((x, y)\) equals \((n*8, 1)\).
• Minimums, or troughs, appear at \(n*8+4, -1\).
• Zeros, where the function crosses the x-axis, appear at \(n*8 + 2, 0\).

The adjustments made due to the factor in the cosine function shift the wave's period, altering the regularity and position of these points in the waveform.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are essential tools in mathematics. They relate the angles and sides of right triangles, but their applications extend to modeling periodic phenomena.

The cosine function, \(\cos(x)\), particularly centers around its periodic nature and symmetrical graph. It oscillates between 1 and -1, creating a wave pattern that repeats endlessly. In transformations like \(y=\cos\left(\frac{\pi x}{4}\right)\), changes in the formula affect:

• The period, dictated by any coefficient of \(x\), which alters the interval of repetition.
• The amplitude, though not altered directly here, impacts wave height in more complex transformations.
• The phase shift, which may occur if horizontal or vertical translations are applied.

Understanding these components helps in analyzing and graphing trigonometric functions correctly to model real-world scenarios effectively.