Problem 26 - Algebra and Trigonometry

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 is 1, period is \(\frac{1}{2}\).

Step 1: Identify the Standard Form

The function is given as \( y = -2 + \cos(4\pi x) \). The standard form of a cosine function is \( y = a + \cos(bx + c) \). Here, we see that \( a = -2 \), \( b = 4\pi \), and \( c = 0 \).

Step 2: Determine the Amplitude

For a cosine function \( y = a + \cos(bx + c) \), the amplitude is given by the coefficient of \( \cos \), which is \( 1 \) in this case. Amplitude is always positive, so the amplitude is 1.

Step 3: Determine the Period

The period of a cosine function is given by the formula \( \frac{2\pi}{b} \). Here, \( b = 4\pi \). So, the period is \( \frac{2\pi}{4\pi} = \frac{1}{2} \).

Step 4: Sketch the Graph

Start by drawing the x-axis and y-axis. The midline of the function is \( y = -2 \), so the graph oscillates above and below this line. The amplitude of 1 means it reaches up to \( y = -1 \) and down to \( y = -3 \). The period is \( \frac{1}{2} \) units, meaning each full cycle of the cosine wave occurs within half a unit on the x-axis.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted as \( \cos \). It is periodic and describes a wave-like pattern characterized by peaks and troughs. Unlike the sine function, the cosine function achieves its maximum value at zero before beginning its descent. In general, the equation for a standard cosine function is given as: \( y = a + \cos(bx + c) \), where \( a \) shifts the graph vertically, \( b \) affects the period of the function, and \( c \) adjusts the phase or horizontal shift. Understanding the parameters of the cosine function helps in determining the shape and orientation of the graph, making it easier to visualize.

Amplitude

Amplitude refers to the height of the peaks in a trigonometric graph or the depth of the troughs below the midline. For a cosine function of the form \( y = a + \cos(bx + c) \), the amplitude is represented by the coefficient in front of the cosine term. However, unlike transformations, it is always taken as a positive number. In the equation \( y = -2 + \cos(4\pi x) \), the amplitude is 1, meaning the graph will oscillate one unit above and below the midline. Knowing the amplitude is crucial as it tells us how far the graph stretches on the y-axis.

Period of a Function

The period of a function describes the length over which a trigonometric graph repeats its pattern. Specifically, for the cosine function \( y = a + \cos(bx + c) \), the period is determined by \( \frac{2\pi}{b} \). In this scenario, where \( b = 4\pi \), the period is calculated as \( \frac{2\pi}{4\pi} = \frac{1}{2} \). This indicates that the cosine wave completes one full cycle every half unit along the x-axis. Knowing the period helps you understand how frequently the graph cycles appear, important for sketching accurate graphs over larger spans.

Graph Sketching

To sketch the graph of a trigonometric function like \( y = -2 + \cos(4\pi x) \), you'll follow a series of steps. First, draw the x-axis and y-axis as a reference framework. Identify the midline, which in this example is \( y = -2 \). The graph will center and oscillate around this line. The amplitude of 1 shows the graph stretches from \( y = -3 \) to \( y = -1 \). Given the period of \( \frac{1}{2} \, \text{units} \), label your x-axis with increments to reflect this, noting that the wave repeats every half unit. Plot key points, such as maximum, minimum, and intercepts, then sketch the wave with smooth, consistent curves to complete the graphing process. The final graph visually demonstrates the behavior described by the mathematical function.