Problem 63 - Precalculus: Functions and Graphs

Question

Sketch the graph of the equation. $$y=2^{-x} \cos x$$

Step-by-Step Solution

Answer

The graph shows oscillations decreasing in amplitude as x increases.

Step 1: Analyze the Components

The given equation is \( y = 2^{-x} \cos x \). This is a product of an exponential function \( 2^{-x} \) and a trigonometric function \( \cos x \). The exponential part, \( 2^{-x} \), represents a decreasing function, whereas \( \cos x \) oscillates between \(-1\) and \(1\). Analyze how these two functions will interact.

Step 2: Identify Key Features

Identify key features of the function. Since \( 2^{-x} \) is always positive and decreasing, it affects the amplitude of the oscillations of \( \cos x \). The amplitude of \( y = 2^{-x} \cos x \) will decrease as \( x \) becomes larger. The zeros of \( \cos x \), where \( \cos x = 0 \), will also be the zeros of the entire function \( y \).

Step 3: Plot Behavior Over an Interval

Let’s consider the interval from \( x = 0 \) to \( x = 2\pi \). Calculate a few points: for \( x=0 \), \( y=1 \); for \( x=\pi/2 \), \( y=0 \); for \( x=\pi \), \( y=-\frac{1}{2^\pi} \); and for \( x=2\pi \), \( y=\frac{1}{2^{2\pi}} \). The exponential part causes the amplitude of the wave-pattern to decrease as \( x \) increases.

Step 4: Sketch the Graph

Sketch the graph using the key points and characteristics identified. Start with the oscillating pattern of \( \cos x \), marking the zeros at \( x=\pi/2, 3\pi/2, 5\pi/2, \dots \). Apply the exponential decay from step 2, shrinking the wave’s amplitude after each period as \( x \) increases.

Step 5: Reflect on Symmetry and Asymptotic Behavior

The function \( y = 2^{-x} \cos x \) is neither symmetric about the y-axis nor the origin. It does approach zero as \( x \to \infty \) due to the exponential decay component \( 2^{-x} \). Keep this asymptotic behavior in mind when finalizing the graph.

Key Concepts

Exponential Decay

Exponential decay is a concept where a quantity decreases at a rate proportional to its current value. In the graph of the function \( y = 2^{-x} \cos x \), the term \( 2^{-x} \) causes this decay. As \( x \) increases, \( 2^{-x} \) becomes smaller, approaching zero. This means the amplitude—how high or low the wave of the cosine function goes—shrinks as \( x \) moves towards positivity.
This behavior can be observed in real-world phenomena, such as cooling substances, radioactive decay, and population decreases. For our function, this property ensures that no matter how long the calculation or graphing extends, the function's effect will lessen over time. Additionally, it's helpful to visualize the exponential base, \( 2 \), dictating how rapidly the function decreases with increasing \( x \).

Amplitude Modulation

Amplitude modulation refers to the process of varying the strength or level of a waveform. Imagine a radio station transmitting signals; the amplitude of the sound waves changes without altering the frequency. In \( y = 2^{-x}\cos x \), the amplitude of \( \cos x \) is controlled by the \( 2^{-x} \) component.
This means that initially, when \( x \) is small, the function behaves just like \( \cos x \), fluctuating between \(-1\) and \(1\). However, as \( x \) grows, the control \( 2^{-x} \) exerts becomes noticeable. It compresses the wave's peaks and troughs, causing a gradual fade of the wave's intensity over time. Hence, as you extend the graph, you'll notice the oscillations become tighter, a visual result of this modulation effect.

Zeroes of a Function

Zeroes of a function are the input values (\( x \)) where the function equals zero. For the equation \( y = 2^{-x} \cos x \), the zeroes of the function occur wherever \( \cos x = 0 \). This happens at points such as \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \).
In practice, these zero points are pivotal—they form the nodes where the wave crosses the x-axis. Identifying these points guides how you sketch the function. Since \( 2^{-x} \) is never zero, the zeros rely entirely on the cosine component. During graphing, these points manifest as contact points of the decreasing wave pattern, punctuating intervals along the x-axis at regular semi-circle distances.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches infinity. In the case of \( y = 2^{-x} \cos x \), the function tends towards zero as \( x \rightarrow \infty \). This occurs because the exponential decay term, \( 2^{-x} \), continuously decreases.
As the graph stretches towards higher\( x \)-values, the height of the wave's crests decreases, becoming imperceptible eventually. While the cosine function endlessly oscillates, the multiplier \( 2^{-x} \) ensures the values approach—but never truly reach—zero. This behavior helps in understanding and predicting long-term tendencies of functions in broader applications, emphasizing their diminishing significance over extended intervals.