Problem 12 - Algebra 2

Question

Write a cosine function for each description. Assume that \(a>0\). amplitude \(\pi,\) period 2

Step-by-Step Solution

Answer

The cosine function for the given description is \(f(x) = \pi \cdot \cos(\pi x)\).

Step 1: Identify Amplitude

The amplitude given in the question is \(\pi\). The amplitude is represented by the coefficient 'a' in the general form of the cosine function. So, a = \(\pi\).

Step 2: Identify Period and Calculate Coefficient b

The period given in the question is 2. The period is inversely proportional to the coefficient 'b' in the general form of cosine function. As per the standard form, when b=1, the period is \(2\pi\). So, if the period is not \(2\pi\), we calculate b as \(b = \frac{2\pi}{\text{period}}\). In this case, the period is 2, so \(b = \frac{2\pi}{2} = \pi\).

Step 3: Write the Function

According to the general form of cosine function \(f(x) = a \cdot \cos(bx)\), replace 'a' and 'b' with their calculated value. So the cosine function is written as \(f(x) = \pi \cdot \cos(\pi x)\).

Key Concepts

Amplitude in Trigonometry

Amplitude is a crucial aspect of trigonometric functions that describes the height of the peaks above and below the midline of the wave. In a cosine function, such as \( f(x) = a \cdot \cos(bx) \), the amplitude is represented by the absolute value of the coefficient 'a'.

It's important to recognize that the amplitude determines how far the function stretches vertically from its central axis. For instance, in the exercise provided, the amplitude is \( \pi \), indicating that from the center (usually the x-axis), the wave will reach heights and depths of \( \pi \).

• The amplitude is always a positive value.
• It measures the distance from the highest to the lowest point of the wave.
• Without any transformations, the cosine function's natural amplitude is 1.

This makes it an integral parameter when considering graphs and transformations of trigonometric functions.

Period of Trigonometric Functions

The period of a trigonometric function is the distance along the x-axis required for the function to complete one full cycle. Understanding how to find and manipulate the period of functions like the cosine function is essential.

In a general cosine function \( f(x) = a \cdot \cos(bx) \), the period is calculated as \( \frac{2\pi}{b} \). This formula stems from the fact that a standard cosine curve repeats every \( 2\pi \) units if \( b = 1 \).

• If \( b > 1 \), the function completes its cycle faster, resulting in a shorter period.
• If \( 0 < b < 1 \), the cycle takes longer, creating a longer period.

In the solution given, with a period of 2, we solve \( \frac{2\pi}{b} = 2 \), resulting in \( b = \pi \). This tells us the function repeats every 2 units along the x-axis, guided by the coefficient 'b'.

Trigonometric Function Transformations

Trigonometric function transformations include changes in amplitude, period, phase shifts, and vertical shifts. These transformations alter the appearance and behavior of the function's graph.

For cosine functions, transformations follow the general form \( f(x) = a \cdot \cos(bx + c) + d \). Here's what each component does:

• **Amplitude (a)**: Controls the vertical stretch or shrink of the graph.
• **Period (b)**: Regulates how quickly the graph repeats its cycle, affecting the period.
• **Phase Shift (c)**: Moves the graph horizontally along the x-axis.
• **Vertical Shift (d)**: Adjusts the graph up or down on the y-axis.

Understanding these transformations helps in sketching the graph accurately and recognizing the function’s characteristics. In our example, the function transformation gave us \( f(x) = \pi \cdot \cos(\pi x) \), showing a change in amplitude and period without phase or vertical shifts.