Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The graph of \( y = e^x \sin x \) is an oscillating, exponentially growing wave with peaks at \( x = n\pi + \pi/2 \) and zeros at \( x = n\pi \) for integers \( n \).

Step 1: Understand the Components

The equation given is a product of two functions: an exponential function, \( y = e^x \), and a trigonometric function, \( y = \sin x \). Understanding how both functions behave on their own will help us combine them. The exponential function \( e^x \) is always positive and increases as \( x \) increases, while the \( \sin x \) function oscillates between -1 and 1.

Step 2: Analyze the Amplitude and Growth

Since \( y = e^x \) is always positive and \( |\sin x| \leq 1 \), the graph of \( y = e^x \sin x \) will have its peaks and troughs determined by the product of a growing exponential and a periodic sine wave. Therefore, the amplitude of the oscillations increases as \( x \) grows positively.

Step 3: Determine Symmetry

The sine function, \( \sin x \), is symmetric, but \( e^x \) is not. Thus, \( y = e^x \sin x \) does not possess any symmetry such as even or odd, making the graph asymmetric about the y-axis or origin.

Step 4: Study the Behavior at Intervals

Observe key points where the sine wave peaks: multiples of \( \pi \) ( such as \( \pm \pi/2, \pm 3\pi/2 \)), and zeros at integer multiples of \( \pi \). The function \( y = e^x \sin x \) will touch the x-axis at these multiples of \( \pi \), since \( \sin x = 0 \).

Step 5: Sketch the Graph

Begin sketching the graph by plotting points where \( x \) is an integer multiple of \( \pi /2 \), noting that the graph rises sharply on the right side of the y-axis and oscillates increasingly, as \( x \to \infty \). The graph to the left, where \( x \to -\infty \), oscillates towards zero since \( e^x \to 0 \). Plot additional points to ensure accuracy.

Key Concepts

Exponential Functions

Exponential functions, like the one in our equation, are a special type of mathematical function where the variable appears in the exponent.
The general form is given by:

• \(y = a^x\), where \(a\) is a constant.

In our specific case, \(y = e^x\), the base \(e\) is an irrational number approximately equal to 2.718.
This function always grows exponentially as \(x\) increases, producing a characteristic curve that starts slowly and rises ever more steeply.
Important characteristics of exponential functions:

• They are always positive (if the base is greater than 1).
• The function grows quickly as \(x\) becomes very large.
• For negative \(x\)-values, the function approaches zero but never reaches it.

Trigonometric Functions

Trigonometric functions, such as \(\sin x\), are fundamental in mathematics, especially in periodic systems.
These functions repeat their values in regular intervals, or cycles, known as their periods.
For \(\sin x\), the function oscillates between -1 and 1 with a period of \(2\pi\).
This means over one full cycle, it traces a wave-like oscillating pattern:

• At \(x = 0\), \(\sin x = 0\).
• Peaks at \(x = \pi/2\) (\(\sin x = 1\)).
• Returns to zero at \(x = \pi\).
• Trough at \(x = 3\pi/2\) (\(\sin x = -1\)).
• Completes the cycle at \(x = 2\pi\).

The beauty of trigonometric functions in physics and engineering lies in their ability to model oscillatory motions and waves.

Graph Sketching

Sketching the graph of complex functions like \(y = e^x \sin x\) can be made easier by analyzing the behavior of each component.
The goal is to combine both the growth rate of the exponential part and the oscillations of the trigonometric part.

• Start by identifying key points: The sine component dictates zeros, peaks, and troughs.
• Plot the zero-crossing points: Multiples of \(\pi\) where \(\sin x = 0\) cause the graph to intersect the \(x\)-axis.
• Note the behavior around peaks/troughs: At these points, multiply the current \(\sin x\) value by the corresponding \(e^x\) value to determine the absolute peaks and troughs in the graph.

Begin connecting these points with smooth curves that mimic the exponential growth and sinusoidal wave, adjusting for the amplitude change as governed by the product of the two functions.

Asymmetry in Graphs

Asymmetry in graphs often arises from the properties of the individual functions involved.
In the equation \(y = e^x \sin x\), we observe that the exponential function \(e^x\) is inherently asymmetric.
While \(\sin x\) is symmetric in itself, its multiplication with \(e^x\) removes any potential symmetry in the combined function.

• \(e^x\) shifts the balance of symmetry because it grows uniquely in positive \(x\)-direction, causing the graph to skew towards positive values as \(x\) increases.
• The lack of an axis of symmetry means the graph is not mirrored across the y-axis nor is it the same on both sides of the origin.

Understanding these characteristics helps in visualizing and predicting the overall shape of such composite graphs.