Problem 66 - Precalculus: Functions and Graphs

Question

Sketch the graph of the equation. $$y=|x| \cos x$$

Step-by-Step Solution

Answer

The graph of \( y = |x| \cos x \) is a symmetrical wave about the y-axis, increasing in amplitude as you move away from the origin.

Step 1: Understand the Equation

The given equation is \( y = |x| \cos x \). This equation combines two functions: the absolute value function \( |x| \) and the cosine function \( \cos x \). It's important to recognize that \( |x| \) affects the amplitude of the cosine wave, making all values non-negative.

Step 2: Identify Key Points

Examine the behavior of \( |x| \) and \( \cos x \). At \( x = 0 \), \( |x| = 0 \), so \( y = 0 \). For other points such as \( x = \pi/2 \), \( 3\pi/2 \), \( 5\pi/2 \), and so on, the value of \( \cos x \) is 0, hence \( y = 0 \) there as well.

Step 3: Determine the Behavior for x > 0

For \( x > 0 \), the function becomes \( y = x \cos x \). Recognize that the cosine function oscillates between -1 and 1, so \( y \) oscillates between \(-x\) and \(x\). The graph resembles a cosine wave increasing in amplitude since \( x \) is increasing for positive values.

Step 4: Determine the Behavior for x < 0

For \( x < 0 \), \( |x| = -x \), so the function is \( y = -x \cos x \). However, since absolute value makes the graph non-negative, it behaves similarly but mirrored about the y-axis as compared to \( x > 0 \).

Step 5: Sketch the Graph

Plot the key points found in Step 2 on a coordinate plane. For positive \( x \), sketch a cosine wave starting at the origin, with increasing amplitude as \( x \) increases. For negative \( x \), mirror this wave across the y-axis. The graph will exhibit a wave-like form, symmetrically growing in amplitude away from the origin.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics, expressed as \(|x|\). It transforms any input value into its non-negative counterpart. This means that whether \(x\) is positive or negative, \(|x|\) will always be positive or zero.

• If \(x > 0\), then \(|x| = x\).
• If \(x = 0\), then \(|x| = 0\).
• If \(x < 0\), then \(|x| = -x\).

In the context of graphing \(y = |x| \cos x\), the absolute value influences the graph by ensuring that all \(y\)-values remain non-negative. This is an important characteristic as it affects how the cosine function's output is transformed. The absolute value effectively mirrors any negative outputs of \(x\) to positive, causing the graph to exhibit certain symmetries around the y-axis.

Cosine Function

The cosine function, denoted as \(\cos x\), is a periodic trigonometric function that oscillates between -1 and 1. Its key feature is its wave-like pattern that repeats every \(2\pi\) radians.

• The cosine of 0 is 1.
• The cosine of \(\pi/2\) is 0.
• The cosine of \(\pi\) is -1.
• It completes one full cycle every \(2\pi\) radians.

In the equation \(y = |x| \cos x\), the cosine function's behavior remains unchanged in its periodicity, but the amplitude is modified by the absolute value function \(|x|\). The turning points of the cosine function at integer multiples of \(\pi/2\) are crucial, as they determine key points on the graph where \(y = 0\). These points help in constructing the wave with increasing amplitude as \(|x|\) increases.

Amplitude of Function

Amplitude refers to the maximum distance a wave reaches from its mean position. For a standard cosine function, the amplitude is typically 1. However, in the equation \(y = |x| \cos x\), the amplitude changes dynamically because of the multiplication by \(|x|\).

• For \(x > 0\), the amplitude is \(x\), causing the wave to grow outward as \(x\) increases.
• For \(x < 0\), the function mirroring keeps the amplitude equivalent to the positive side.

The increase in amplitude with \(|x|\) implies that the wave's peaks and troughs grow further from the center line as we move away from the origin in either direction. This is a unique behavior resulting from the interaction between the absolute value and cosine functions.

Symmetry in Graphs

Symmetry in graphs refers to the balanced and mirrored appearance of a graph around a certain line or point. In the equation \(y = |x| \cos x\), symmetry is particularly noteworthy. The involvement of the absolute value function results in a graph that is symmetric about the y-axis.

• For \(x > 0\), the graph takes the form of a typical positive cosine wave with increasing amplitude.
• The negative values of \(x\), due to \(|x|\), mirror this positive cosine wave across the y-axis.

This symmetry is a direct consequence of the absolute value function folding any negative outputs upwards, which visually duplicates the positive side of the graph for the negative side, creating a harmonious and predictable pattern.