Problem 20 - Precalculus

Question

In Exercises \(11-20,\) state the amplitude and period of each function. $$y=4 \cos \left(\frac{\pi}{4} x\right)$$

Step-by-Step Solution

Answer

Amplitude is 4 and period is 8.

Step 1: Identify the General Form

Recognize that the function is of the form \( y = a \cos(bx + c) + d \), where \( a = 4 \), \( b = \frac{\pi}{4} \), and both \( c \) and \( d \) are 0. This will help us determine both the amplitude and period of the function.

Step 2: Find the Amplitude

The amplitude of a cosine function \( y = a \cos(bx + c) + d \) is given by the absolute value of \( a \). Hence, the amplitude of the function \( y = 4 \cos\left(\frac{\pi}{4} x\right) \) is \( |4| = 4 \).

Step 3: Calculate the Period

The period of a cosine function depends on the coefficient \( b \) and is calculated using the formula \( \frac{2\pi}{b} \). For the function \( y = 4 \cos\left(\frac{\pi}{4} x\right) \), the period is \( \frac{2\pi}{\frac{\pi}{4}} = 8 \).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that is useful in various mathematical, physical, and engineering applications. When you hear "cosine," think of a wave that represents how an angle changes in a circle as the angle itself increases. This function is often expressed as \( y = \cos(x) \) and is known for its repeated, wave-like pattern.

• Wave Nature: The cosine function creates a wave that starts at its maximum value, dips to a minimum, and returns to the maximum in predictable cycles.
• Normal Range: The values that \( \cos(x) \) can take are between -1 and 1, bouncing up and down perfectly like the undulations of a wave.
• General Form: In its more generalized form, \( y = a \cos(bx + c) + d \), changing \(a\) affects the height of the peaks (amplitude), while \(b\) impacts the distance between these peaks (period).

The example function \( y = 4 \cosigg(\frac{\pi}{4}x\bigg) \) has an 'a' value of 4, stretching the wave vertically, making the peaks and troughs four times as high and deep compared to the basic cosine wave.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry, essential for understanding relationships in right-angled triangles and modeling periodic phenomena. The six main trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant.

• Key Relationships: These functions relate the angles of a triangle to the lengths of its sides. They are defined for all types of angles and can describe circular motion as well.
• Cosine's Role: Cosine, specifically, links the adjacent side and hypotenuse of a right triangle. It is cyclic and repetitive, making it perfect for representing cycles like day and night, waves, and circular motions.

What makes cosine special in our context is its ability to describe how things oscillate over time. It's used widely to model systems that are periodic in nature such as sound waves, alternating current, and tides.

Periodicity in Functions

Periodicity is a property of functions where values repeat at regular intervals, creating a consistent pattern over time. An understanding of periodicity is crucial when dealing with any oscillating or cyclic system.

• Identifying Periodicity: In a function like \( y = a \cos(bx + c) + d \), the period is determined by the variable \( b \), with the formula \( \frac{2\pi}{b} \) indicating how long it takes for the function to complete one full cycle.
• Application of Periodic Functions: Phenomena such as sound waves and seasonal changes can be characterized with functions displaying distinct periodic behavior.

For \( y = 4 \cos\left(\frac{\pi}{4} x\right) \), with \( b = \frac{\pi}{4} \), the periodicity is calculated as \( \frac{2\pi}{\frac{\pi}{4}} = 8 \), meaning every 8 units along the x-axis, the function's pattern repeats exactly, showcasing the complete cycle of its wave.