Problem 28 - Precalculus Mathematics for Calculus

Question

Find the amplitude and period of the function, and sketch its graph. $$y=-2+\cos 4 \pi x$$

Step-by-Step Solution

Answer

Amplitude: 1; Period: \( \frac{1}{2} \).

Step 1: Understanding the Basic Form

The cosine function can be expressed in the form \( y = a + b \cos(c x + d) \). Here, \( a = -2 \), \( b = 1 \), \( c = 4\pi \), and \( d = 0 \). This understanding tells us the effects on amplitude and translation for the function.

Step 2: Finding the Amplitude

The amplitude of a function \( y = a + b \cos(c x + d) \) is given by the absolute value of \( b \), which is \( |b| = |1| = 1 \). This means the graph oscillates 1 unit above and below the center position, which in this case is shifted to \( y = -2 \).

Step 3: Calculating the Period

The period of the function is calculated using the formula \( \frac{2\pi}{|c|} \). Plugging in \( c = 4\pi \), the period is \( \frac{2\pi}{4\pi} = \frac{1}{2} \). Therefore, the function completes one cycle in an interval of \( \frac{1}{2} \) on the \( x \)-axis.

Step 4: Sketching the Graph

To sketch the graph, start by drawing the midline at \( y = -2 \). From here, the graph will oscillate between \( y = -1 \) and \( y = -3 \). Since the period is \( \frac{1}{2} \), the pattern repeats every \( \frac{1}{2} \) units along the \( x \)-axis. The graph starts at the maximum point (as \( \cos(0) = 1 \)), at \( (0, -1) \), and cycles through its usual cosine shape within the period.

Key Concepts

Amplitude

In trigonometry, the concept of "amplitude" refers to the height of the wave from its central axis or midline. It defines how far the wave oscillates from its mean value. For any given cosine or sine function represented as \( y = a + b \cos(c x + d) \), the amplitude is determined by the coefficient \( b \). It is important to note that this value is always positive, as it represents a distance. Therefore, we take the absolute value of \( b \), expressed mathematically as \(|b|\).

• For example, in the function \( y = -2 + \cos(4\pi x) \), the amplitude is \(|1| = 1\).
• This indicates the wave oscillates 1 unit above and below the midline.
• The midline itself is shifted to \( y = -2 \).

Remember, amplitude doesn’t affect the frequency or period of the wave but solely determines how tall the wave appears vertically.

Period of a Function

The "period" of a trigonometric function is an essential concept that describes how long it takes for the wave to repeat its pattern. For a cosine function in the form \( y = a + b \cos(c x + d) \), the period is calculated using the formula \[\text{Period} = \frac{2\pi}{|c|}.\]

• In the given function \( y = -2 + \cos(4\pi x) \), \( c = 4\pi \).
• By substituting \( c \) into the formula, the period becomes \( \frac{2\pi}{4\pi} = \frac{1}{2} \).
• This result means the waveform completes a full cycle every \( \frac{1}{2} \) units along the \( x \)-axis.

Understanding the period is crucial because it allows you to sketch the function's graph accurately over the correct interval. The shorter the period, the more cycles the graph completes over a given stretch of \( x \). Conversely, a longer period means fewer cycles over the same distance.

Graphing Trigonometric Functions

Graphing trigonometric functions such as sine and cosine involves understanding their amplitude, period, phase shifts, and vertical shifts. In our example function \( y = -2 + \cos(4\pi x) \), several steps guide you through sketching the graph:

• Start with the midline: For this function, the midline is at \( y = -2 \). This line serves as the baseline from which the wave oscillates.
• Determine amplitude alterations: With an amplitude of 1, the wave will rise to \( y = -1 \) and fall to \( y = -3 \).
• Calculate the period: Knowing the period is \( \frac{1}{2} \), it shows the cycle completes every half a unit on the \( x \)-axis.
• Plot key points: The graph begins at its maximum at \( (0, -1) \), dips down to the minimum at \( (\frac{1}{4}, -3) \), and returns to maximum at \( (\frac{1}{2}, -1) \).
• Connect the wave: Tracing these points through a smooth curve will give you the characteristic cosine wave shape repeated every half unit.

Visualizing trigonometric functions involves plotting several cycles to see how the midline, amplitude, and period bring together the sine or cosine wave shape.