Problem 21 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The graph of \( y = \cos(\pi x) \) is a cosine wave with a period of 2, amplitude of 1, crossing the x-axis at multiples of 0.5 and repeating every 2 units.

Step 1: Understand the Function

First, recognize that the function given is a cosine function where the angle is multiplied by \( \pi \), specifically \( y = \cos(\pi x) \). This suggests that the standard cosine function \( \cos(\theta) \) is being modified by \( \pi x \), which affects the graph's period.

Step 2: Identify the Period

The standard period of \( \cos(\theta) \) is \( 2\pi \). Since our function is \( \cos(\pi x) \), calculate the period as follows: period = \( \frac{2\pi}{\pi} = 2 \). Thus, the function completes one full cycle every 2 units along the x-axis.

Step 3: Determine Amplitude and Vertical Shift

The standard cosine function has an amplitude of 1 and no vertical shift. Therefore, \( y = \cos(\pi x) \) also has an amplitude of 1 and is centered around the x-axis, i.e., the horizontal line \( y = 0 \).

Step 4: Identify Key Points

Determine key points based on the identified period and amplitude: at \( x=0, 1, 2 \), \( y = \cos(\pi x) \) yields \( \cos(0) = 1 \), \( \cos(\pi) = -1 \), and \( \cos(2\pi) = 1 \). Thus, starting at \( x=0 \), the pattern repeats every 2 units.

Step 5: Plot the Key Points and Sketch the Curve

Using the key points \( (0, 1), (1, -1), (2, 1) \), plot them on a coordinate grid. As the function is periodic with a period of 2, extend this pattern to draw a smooth and continuous wavy line, repeating these heights and depths for adjacent intervals.

Key Concepts

Period of Trigonometric Functions

In trigonometry, the period of a function refers to the distance over which the function repeats its pattern. For standard trigonometric functions like sine and cosine, the period is typically defined in terms of radians, with the cosine function having a default period of \(2\pi\). This means \(\cos(\theta)\) completes one full wave or cycle every \(2\pi\) units along the x-axis.

When adjustments are made to the function, such as multiplying the angle by a coefficient, the period changes. In the case of the function \(y = \cos(\pi x)\), the input \(x\) is multiplied by \(\pi\). To find the new period, use the formula:

• Period \(= \frac{2\pi}{\text{coefficient of } x}\)

For \(y = \cos(\pi x)\), this calculation becomes \(\frac{2\pi}{\pi} = 2\).

This tells us that the cosine pattern repeats every 2 units along the x-axis. This reduced period results in the graph appearing more frequently over a given interval compared to the standard cosine graph.

Amplitude of Trigonometric Functions

Amplitude describes the height of a wave measured from the centerline to its peak. For simple harmonic functions like the cosine, it often refers to the maximum displacement from the horizontal axis.
Typically, the cosine function \(\cos(\theta)\) has an amplitude of 1, which means its highest point is 1 unit away from the axis and the lowest is -1 unit.

When examining \(y = \cos(\pi x)\), there's no vertical stretch or compression mentioned, meaning the amplitude remains 1. Thus, regardless of the period adjustment, this function maintains its standard amplitude, reaching a peak of 1 and a trough of -1. The amplitude gives you a visual guide of how tall the wave appears on a graph.

Cosine Function

The cosine function \(y = \cos(x)\) is one of the core building blocks in trigonometry. It's a smooth, continuous wave that oscillates between -1 and 1.

Understanding the cosine graph involves knowing its key characteristics:

• Period: As discussed, the default period is \(2\pi\), but for functions modified like \(y = \cos(\pi x)\), the period is adjusted based on the formula we apply.
• Amplitude: This is the fixed height of the wave from its center, and for cosine, it's usually 1 unless otherwise altered by multiplication.
• Waveform: Cosine starts at its peak when \(x = 0\), moving to its trough at halfway through the period, and returning to the peak by the period's end.

For \(y = \cos(\pi x)\), the graph begins at the top of its wave at 1 when \(x=0\), descends to -1 at \(x=1\), and rises back to 1 when \(x=2\).

This distinctive wave continues indefinitely, capable of complex interactions when combined with other functions but always maintaining its foundational properties.