Problem 20 - Calculus for Biology and Medicine

Question

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=0.2 \cos (-x) $$

Step-by-Step Solution

Answer

The graph is a cosine wave with amplitude 0.2, reflected over the y-axis, and period \(2\pi\).

Step 1: Identify the Basic Function

The given function is based on the cosine function, which is a periodic trigonometric function. The basic cosine function is defined as \( y = \cos(x) \).

Step 2: Account for the Transformation by -x

The function \( y = 0.2 \cos(-x) \) has a phase reflection. The negative sign inside the cosine argument reflects the graph across the y-axis. For the cosine function, this reflection does not change the appearance of the graph.

Step 3: Identify the Amplitude Change

The coefficient 0.2 in front of the \( \cos(-x) \) affects the amplitude of the wave. The amplitude is normally 1, but multiplying by 0.2 scales all y-values by 0.2, making the amplitude 0.2.

Step 4: Determine the Period

The period of the cosine function \( y = \cos(x) \) is \( 2\pi \). Since there are no horizontal transformations other than reflection, the period remains \( 2\pi \).

Step 5: Sketch the Graph

To sketch \( y = 0.2 \cos(-x) \), start by plotting key points based on the reflection and amplitude: \( (0, 0.2) \), \( (\pi/2, 0) \), \( (\pi, -0.2) \), \( (3\pi/2, 0) \), \( (2\pi, 0.2) \). Connect these points smoothly, reflecting the periodic nature with the defined period \( 2\pi \) and amplitude 0.2.

Key Concepts

Cosine Function

The cosine function is a fundamental function in trigonometry. Its formula is expressed as \( y = \cos(x) \). This function produces a smooth, repeating wave that oscillates between -1 and 1. The graph begins at its maximum value of 1 when \( x = 0 \), goes down to -1 at \( x = \pi \), and then returns to 1 at \( x = 2\pi \). The shape of the cosine wave is distinctive, with a peak, a trough, and again a peak, resembling a sinusoidal shape.

In practical scenarios, the cosine function models periodic phenomena such as sound waves, light waves, and tides. Its characteristic shape and behavior make it invaluable for representing cyclic patterns. The inherent symmetry about the y-axis of the cosine function leads it to have even properties, meaning \( \cos(-x) = \cos(x) \). This aspect often simplifies problems involving refelction across an axis.

Amplitude

Amplitude is a term that describes the maximum extent of a wave measured from its equilibrium position. In the context of the cosine function, the amplitude is the absolute highest value the function reaches from the mid-line of the wave.

For a standard cosine function, \( y = \cos(x) \), the amplitude is 1. However, when the function is multiplied by a coefficient, as in \( y = 0.2\cos(x) \), the amplitude becomes 0.2. This means the wave reaches a maximum of 0.2 and a minimum of -0.2, shrinking the wave vertically.

• This stretching or compressing does not affect the wave's periodic properties.
• The mid-point or equilibrium position of the cosine graph remains at the x-axis or \( y = 0 \).

Understanding amplitude is crucial for accurately sketching or understanding physical phenomena, especially when identifying waves' energy or intensity in fields such as physics and engineering.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions over a consistent interval. For the cosine function, the period is the length of one complete cycle of the wave.

For \( y = \cos(x) \), the period is \( 2\pi \). This means, after an input of \( 2\pi \) radians, the function starts over, creating identical peaks and troughs as before. This periodic repetition defines the predictability and consistency of the wave.
The concept of periodicity helps:

• Predict future values of the function beyond the initially observed interval.
• Simplify calculations in cycles, as the values repeat after each period.
• Analyze complex wave superpositions in physics and engineering tasks.

In transformations, such as \( y = 0.2\cos(-x) \), the basic periodicity of \( 2\pi \) is maintained despite changes to amplitude or reflections. Therefore, periodicity remains a key element in function transformation analysis.