Problem 64 - Precalculus: Functions and Graphs

Question

Sketch the graph of the equation. $$y=e^{x} \sin x$$

Step-by-Step Solution

Answer

The graph of \( y = e^x \sin x \) oscillates between \(-e^x\) and \(e^x\), increasing in amplitude as \(x\) increases.

Step 1: Understand the Components of the Equation

The given equation is \( y = e^x \sin x \). It is a product of two functions: an exponential function \( e^x \) and a trigonometric function \( \sin x \). Understanding the behavior of both functions is crucial for sketching the graph.

Step 2: Analyze the Exponential Component

The function \( e^x \) is an exponential function which grows rapidly as \( x \) increases. It is always positive, never zero, and increases without bound as \( x \) approaches infinity. As \( x \) approaches negative infinity, \( e^x \) approaches zero.

Step 3: Analyze the Sine Component

The function \( \sin x \) is a periodic trigonometric function, oscillating between -1 and 1 with a period of \( 2\pi \). This function contributes oscillations to \( y = e^x \sin x \), causing the overall function to fluctuate.

Step 4: Determine Key Points and Intersections

Determine key points where you might expect intersections with axes or symmetry. Notably, when \( x = 0 \), \( y = e^0 \sin 0 = 0 \). Consider points where \( \sin x = 1 \) or \( \sin x = -1 \) as these will determine the oscillatory behavior.

Step 5: Sketch the Behavior of the Function

The graph will oscillate between \( -e^x \) and \( e^x \) because \( \sin x \) oscillates between -1 and 1. As \( x \) increases, the oscillations of the function increase in amplitude due to the exponential growth of \( e^x \). As \( x \) decreases, the oscillations approach zero because \( e^x \) approaches zero.

Step 6: Identify Asymptotic Behavior

As \( x \to \infty \), the function grows larger due to the exponential part \( e^x \). As \( x \to -\infty \), the whole function approaches zero because of the dominance of the exponential decay.

Step 7: Finalize the Sketch

Plot the critical points and oscillatory behavior. The amplitude of oscillations should increase on the positive x-axis and decrease on the negative x-axis. This forms an increasing wave-like pattern of the graph.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. A common example of an exponential function is \( e^x \), where \( e \approx 2.718 \) is Euler's number, a fundamental mathematical constant.
Exponential functions like \( e^x \) show some unique characteristics:

• Rapid growth or decay: As \( x \) increases, \( e^x \) grows extremely fast. Conversely, as \( x \) decreases, \( e^x \) approaches zero.
• Always positive: The value of \( e^x \) is always greater than zero, regardless of the value of \( x \).
• Lack of bounds: As \( x \) goes to positive infinity, \( e^x \) increases without bound.

In applications, exponential functions are used to model scenarios involving continuous growth or decay, such as population growth or radioactive decay.

Trigonometric Function

Trigonometric functions explain relationships between the angles and sides of triangles. The sine function, \( \sin x \), is a fundamental trigonometric function that defines a periodic oscillating behavior.
Key aspects of \( \sin x \) include:

• Periodic nature: \( \sin x \) repeats its values every \( 2\pi \), known as the period.
• Range: It oscillates between -1 and 1, with smooth transitions through these values.
• Important zeros: The function equals zero at integer multiples of \( \pi \), that is, \( x = n\pi \), where \( n \) is an integer.

Trigonometric functions like \( \sin x \) are employed in various domains, from physics for wave behavior to engineering for signal processing.

Graph Sketching

Graph sketching involves plotting a function to visually interpret its behavior. For the composition \( y = e^x \sin x \), graph sketching provides a way to examine how the exponential and trigonometric parts interact.
Follow these principles when sketching graphs:

• Determine key points: For \( y = e^x \sin x \), assess where \( \sin x = 0, 1, \text{or} -1 \).
• Check for symmetry: Understanding whether the graph is symmetric can help simplify the sketching process.
• Visualize oscillations: Consider how one function, like \( \sin x \), modulates the other, such as \( e^x \), in terms of amplitude and frequency.

Effective graph sketching helps in comprehending complex mathematical relationships and predicting function behaviors without needing exact calculations.

Periodic Functions

Periodic functions repeat their values in regular intervals or periods. A classic example is the sine function, \( \sin x \), which repeats every \( 2\pi \). This property makes these functions extremely useful for representing cyclical patterns.
Features of periodic functions include:

• Defined period: The period of a function like \( \sin x \) is \( 2\pi \), meaning it completes one full cycle over this interval.
• Predictable repeats: Because the function's behavior repeats, they are ideal for modeling phenomena like sound waves or seasonal temperatures.
• Amplitude adjustments: In combined functions, periodic aspects can adjust the amplitude of a non-periodic component.

Understanding periodic functions aids in dissecting complex graphs like \( y = e^x \sin x \), as it helps reveal how different function components contribute to the overall shape.