Problem 22 - Calculus for Biology and Medicine

Question

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-2 \cos (\pi x / 4) $$

Step-by-Step Solution

Answer

The graph is a cosine wave with an amplitude of 2, period of 8, and reflects across the x-axis.

Step 1: Identify the Amplitude

The function given is \( y = -2 \cos(\frac{\pi}{4}x) \). The coefficient of the cosine function is -2, which represents the amplitude. Since amplitude is always positive, it is 2.

Step 2: Determine the Period

The period can be found using the formula \( \frac{2\pi}{\text{coefficient of } x} \). Here, the coefficient of \( x \) inside the cosine function is \( \frac{\pi}{4} \). So, the period is \[ \frac{2\pi}{\frac{\pi}{4}} = 8. \] This means the function completes one full cycle every 8 units.

Step 3: Identify the Phase Shift

The given function \( y = -2\cos\left(\frac{\pi}{4}x\right) \) does not have any addition or subtraction within the parentheses affecting \( x \). Therefore, there is no phase shift.

Step 4: Determine the Vertical Shift

There is no constant term added or subtracted to the function, so there is no vertical shift. The midline of the graph will be at \( y = 0 \).

Step 5: Plot Key Points and Sketch

For \( y = -2 \cos(\frac{\pi}{4}x) \), start plotting key points over one period from 0 to 8. Key points of cosine typically occur at critical values:- At \( x = 0 \), \( y = -2 \times 1 = -2 \).- At \( x = 2 \), \( y = -2 \times 0 = 0 \).- At \( x = 4 \), \( y = -2 \times -1 = 2 \).- At \( x = 6 \), \( y = -2 \times 0 = 0 \).- At \( x = 8 \), \( y = -2 \times 1 = -2 \).Join these points with a smooth curve to complete the sketch for one period.

Key Concepts

Understanding Amplitude

The amplitude of a trigonometric function measures the height of its wave from the center line to its peak. In the function \( y = -2 \cos\left(\frac{\pi}{4}x\right) \), the amplitude is given by the absolute value of the coefficient in front of the cosine function. For this equation, the coefficient is -2. Hence, the amplitude is \( | -2 | = 2 \). This means that the highest point of the wave is 2 units above the center line, and the lowest point is 2 units below the center line.

To understand amplitude more deeply, it is key to remember:

• The amplitude specifically affects the vertical stretch or compression of the wave.
• Regardless of whether the coefficient is positive or negative, the amplitude itself is always positive as it is a measure of distance.
• A negative coefficient, like -2, does not affect the amplitude calculation but indicates a reflection across the horizontal axis.

Recognizing amplitude is crucial in sketching trigonometric graphs accurately. It helps set the scale for how "tall" or "short" the function appears on the graph.

Determining the Period

The period of a trigonometric function is the distance required for the function to complete one full cycle of its waveform. For a cosine function, the formula to find the period is \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the function.

In our example, \( y = -2 \cos\left(\frac{\pi}{4}x\right) \), the coefficient \( b \) equals \( \frac{\pi}{4} \). Plugging this into the formula, we calculate the period as:

• \( \frac{2\pi}{\frac{\pi}{4}} = 8 \).

This indicates that the cosine function completes one full wave over an interval of 8 units on the \( x \)-axis. Understanding period is crucial because:

• The period dictates how "stretched" or "compressed" the wave appears horizontally.
• A smaller period results in more oscillations in a given space, whereas a larger period results in fewer oscillations.

Knowing the period allows you to evenly distribute critical points across the graph for accuracy.

Characteristics of the Cosine Function

The cosine function is one of the primary trigonometric functions, recognized for its wave-like, repeating pattern. It is fundamental in understanding periodic phenomena.

Here are some characteristics and tips for graphing the cosine function, using \( y = -2 \cos\left(\frac{\pi}{4}x\right) \) as an example:

• Standard Form: The typical form of a cosine function is \( y = a \cos(bx + c) + d \), where \( a \) affects amplitude, \( b \) affects period, \( c \) affects phase shift, and \( d \) affects vertical shift.
• Key Points: The cosine function starts at its maximum point \((a)\), goes to zero, reaches its minimum point \((-a)\), returns to zero, and finishes one full cycle back at its maximum.
• Symmetry: Cosine functions are symmetric about the vertical axis (even function), making them easier to reflect and predict points.
• Reflection: A negative sign before the coefficient, as in \(-2\), reflects the wave across the \( x \)-axis, flipping it up-side down.

Understanding these aspects allows for efficient and accurate graphing without needing a calculator.