Problem 27 - Precalculus with Calculus Previews

Question

Find the amplitude and period of the given function. Sketch at least one cycle of the graph. $$ y=-3 \cos 2 \pi x $$

Step-by-Step Solution

Answer

Amplitude: 3, Period: 1. The graph starts at \( y = -3 \) and completes a cycle by \( x = 1 \).

Step 1: Identify the General Form of the Function

The given function is \( y = -3 \cos(2\pi x) \). This function follows the form \( y = a \cos(bx) \). Here, the amplitude is \( |a| \) and the period is \( \frac{2\pi}{b} \).

Step 2: Find the Amplitude

The amplitude of the function \( y = -3 \cos(2\pi x) \) can be identified as \( |a| = |-3| = 3 \). Thus, the amplitude is 3.

Step 3: Calculate the Period

For the given function, \( b = 2\pi \). The period is calculated as \( \frac{2\pi}{b} = \frac{2\pi}{2\pi} = 1 \). So, the period of the function is 1.

Step 4: Sketch the Function

To sketch the function \( y = -3 \cos(2\pi x) \), begin by noting that it has an amplitude of 3 and period of 1. Starting from \( x = 0 \), one complete cycle occurs by \( x = 1 \). At \( x = 0 \), the cosine function is at maximum, but here it is flipped due to the negative sign, starting at \( y = -3 \). The graph will then move to \( y = 0 \) at \( x = 0.25 \), \( y = 3 \) at \( x = 0.5 \), back to \( y = 0 \) at \( x = 0.75 \), and completes at \( y = -3 \) when \( x = 1 \).

Key Concepts

Understanding Amplitude in Trigonometric Functions

Amplitude is a key feature of trigonometric functions, especially sine and cosine functions. It tells us how "tall" or "short" the waves of the function are compared to the horizontal axis. In simpler terms, amplitude measures how far the peaks or troughs of the graph extend from the average (or rest) position.

For functions like cosine and sine, the formula for amplitude is absolutely centered around the coefficient in front of the trigonometric term. Mathematically, it's expressed as:

• If the function is given as \( y = a \cos(bx) \) or \( y = a \sin(bx) \), then the amplitude \( = |a| \).

In the given function, \( y = -3 \cos(2\pi x) \), \( a = -3 \). Therefore, the amplitude \( = |-3| = 3 \).

So regardless of whether the cosine function vibrates above or below the x-axis, its maximum height or depth is 3 units. The negative sign in \(-3\) simply reflects the graph over the x-axis.

Understanding Periodicity in Trigonometric Functions

Periodicity of a function relates to how often the function repeats itself over the x-axis. In trigonometric terms, it's how long it takes for the wave-like graph to complete one full cycle and start again.

The general formula for the period in trigonometric functions such as sine and cosine is given by:

• For \( y = a \cos(bx) \) or \( y = a \sin(bx) \), the period \( = \frac{2\pi}{b} \).

In this exercise, given \( y = -3 \cos(2\pi x) \), \( b \) is \( 2\pi \). Therefore, the period \( = \frac{2\pi}{2\pi} = 1 \).

This means the graph of this function will complete one complete cycle every 1 unit along the x-axis. It's important to realize that finding the period helps in predicting the behavior of the function consistently over repeated intervals.

Graphing Trigonometric Functions Effectively

Graphing trigonometric functions can seem complicated at first, but becomes much simpler once you break it down to its core components: amplitude, period, and phase shifts. These elements describe the shifts and stretches of the function in relation to its basic template.

To effectively graph a cosine function, follow these steps:

• Identify the amplitude: In this case, \( amplitude = 3 \). This tells us the graph stretches 3 units above and below its midline.
• Determine the period: From \( y = -3 \cos(2\pi x) \), we've calculated that \( period = 1 \). Therefore, the function completes one full wave-season every 1 unit interval on the x-axis.
• Plot key points: Start at \( x=0 \), where the function begins its cycle. Because the amplitude is negative, the graph starts at the lowest point \( y = -3 \). At \( x = 0.25 \), the graph reaches the midline at \( y = 0 \); at \( x = 0.5 \), the highest point \( y = 3 \); and returns through these points until it finishes the cycle back at \( y = -3 \).

Graph these points smoothly while recognizing the cyclic pattern. This method clearly constructs the wave's rhythm and gets you comfortable with sketching trigonometric graphs. By knowing these features, each graph can reveal the dynamic nature of sine or cosine functions.