Problem 68 - 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. $$y=5 \cos (\pi-2 x)+6$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=5 \cos (\pi-2 x)+6\) is 5 and the period is \(\pi\).

Step 1: Identify the amplitude

The amplitude is the absolute value of the coefficient on the cosine function, which in this case is 5. Therefore, the amplitude of the function is 5.

Step 2: Identify the period

The period of the cosine function is calculated as \(\frac{2\pi}{|B|}\). Here, \(B = -2\), so the period is \(\frac{2\pi}{2}\), which simplifies to \(\pi\). Therefore, the period of the function is \(\pi\).

Step 3: Graphing the function

Using a graphing utility (like a graphing calculator, a computer program, or an online tool), the function \(y=5 \cos (\pi-2 x)+6\) can be graphed. Remember to include two full periods in the graph. The amplitude and period identified in the previous steps can help in setting up the appropriate scale on the y-axis and x-axis respectively.

Key Concepts

Amplitude of Cosine Function

When working with trigonometric functions such as cosine, understanding the amplitude is crucial. The amplitude of a cosine function determines the height of the wave from its midline. In simple terms, it is how far the highest point of the wave is from the central line of the graph.

For the function \( y = 5 \cos(\pi - 2x) + 6 \), the amplitude is represented by the absolute value of the coefficient in front of the cosine term. Here, this coefficient is 5, so the amplitude is \( |5| = 5 \).

This means that the maximum value of the function above the midline is 5 units, and the minimum value below the midline is also 5 units. The midline itself is shifted up by 6 units due to the plus 6 added to the original cosine function.

• Amplitude gives the vertical stretch of the graph.
• A larger amplitude means taller waves; a smaller amplitude means shorter waves.

Period of Cosine Function

The period of a cosine function tells us the horizontal length required for the function to complete one full cycle on the graph. In simpler terms, it is the distance between two consecutive peaks or troughs of the wave.

For the equation \( y = 5 \cos(\pi - 2x) + 6 \), the period is determined by the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \). Here, \( B = -2 \). Consequently, the period is \( \frac{2\pi}{|-2|} = \pi \). This result signifies that a complete wave cycle occurs every \( \pi \) units along the x-axis.

• A smaller period results in a more frequent recurring wave pattern.
• A larger period results in a less frequent wave pattern.

Graphing Utilities

Graphing utilities are incredibly useful tools when visualizing complex trigonometric functions. They allow us to quickly and accurately graph functions to understand their key features such as amplitude and period.

With functions like \( y = 5 \cos(\pi - 2x) + 6 \), a graphing calculator or software can create precise graphs. By setting an appropriate scale based on calculated amplitude and period, the utility helps visualize two full cycles of the function efficiently.

• Graphing tools can adjust the scale to account for amplitude, making it easier to see the graph's full range.
• They adjust the x-axis scale for the period, ensuring two full wave cycles are visible when requested.

Remember to input values accurately and double-check setup settings to ensure the graph represents the true mathematical behavior of the function.