Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-10 \cos \frac{\pi x}{6}$$

Step-by-Step Solution

Answer

The amplitude of the graph is 10 and the period of the graph is 12. This is confirmed by graphing, where the wave pattern starts at -10, peaks at 10, and ends at -10 again, repeating this pattern every 12 units for two full periods.

Step 1: Determining the Amplitude

The amplitude is achieved by getting the absolute value of the coefficient of the cosine function. In this problem, the coefficient is -10, therefore the amplitude is |\(-10\)| which equals \(10\).

Step 2: Finding the Period

The period can be found by using the formula \(2\pi\) divided by the absolute value of the coefficient of \(x\). In the function proposed, the coefficient of \(x\) is \(\frac{\pi}{6}\). Thus, the period is \(\frac{2\pi}{\frac{\pi}{6}}\) which simplifies to \(12\).

Step 3: Graphing the Function

A graphing utility can be utilized to graph the function. Upon plotting, it will reveal a wave pattern starting at -10 (since \(\cos{0} = 1\) and the coefficient is -10), then rising to 10, and then back to -10 (completing one period) before starting the next period. The function will be verified to complete two full periods in an interval of 24 units, in line with the calculated period. Note that the graph is inverted due to the negative sign in front of the cosine function.

Key Concepts

Amplitude of Trigonometric Functions

Understanding the amplitude of trigonometric functions is crucial when analyzing waves and vibrations. The amplitude refers to the distance from the center line of the wave to the peak or trough. For a function such as \(y = A \times \text{cos}(Bx + C) + D\), the amplitude is given by the absolute value of coefficient \(A\). This defines how 'tall' or 'short' the waves of the function are.

Considering the given exercise, \(y = -10 \text{cos} \frac{\text{π}x}{6}\), the coefficient of the cosine function is -10. Despite the negative sign, the amplitude is simply the absolute value, resulting in an amplitude of 10. This tells you that the wave's peaks and troughs are 10 units from the center line—regardless of the fact that the wave may start at a peak or a trough due to the negative coefficient.

Period of Trigonometric Functions

The period of a trigonometric function, such as sine or cosine, is the length over which the function repeats. This is a fundamental concept in both mathematics and the physical sciences, where periodic phenomena occur.

For a function \(y = A \text{cos}(Bx)\), the period is calculated using the formula \(\frac{2π}{|B|}\). In this formula, \(B\) signifies how many times the function will repeat over the interval of \(2π\).

In the provided exercise, the period is obtained from \(\frac{2π}{\left| \frac{π}{6} \right|}= 12\). This means that every 12 units along the x-axis, the wave pattern of the given cosine function repeats itself. The knowledge of the period is essential for plotting the graph accurately over any range.

Graphing Utilities in Mathematics

Graphing utilities are incredibly useful tools for visualizing mathematical functions and gaining insights into their behavior. They help take equations from an abstract form and turn them into visual graphs which are much easier to interpret.

Using such tools, students can plot functions like \(y = -10 \text{cos} \frac{πx}{6}\), to observe characteristics such as amplitude and period directly. They can zoom in and out, adjust the scale, and even animate the graph to better understand the dynamic nature of trigonometric functions. This interactive approach enriches the learning experience and makes it easier to comprehend complex concepts.

Properties of the Cosine Function

The cosine function, one of the basic trigonometric functions, is known for its wave-like pattern. Some of its properties include starting at its maximum value when \(x = 0\) if it's not affected by a phase shift, and typically oscillating between 1 and -1 for the standard \(cos(x)\) function.

The given function, \(y=-10 \text{cos} \frac{\text{π}x}{6}\), is a transformed cosine function which has been vertically stretched by a factor of 10, given its amplitude of 10, and horizontally stretched to have a period of 12 units. Another interesting property is that the that due to the negative sign in front of the cosine function, this causes a reflection across the x-axis, resulting in an inverted graph compared to the standard cosine wave.