Problem 63 - Precalculus Mathematics for Calculus

Question

Graph the three functions on a common screen. How are the graphs related? $$y=\sqrt{x}, \quad y=-\sqrt{x}, \quad y=\sqrt{x} \sin 5 \pi x$$

Step-by-Step Solution

Answer

The graphs are related through vertical reflections and sinusoidal modulation of \( y=\sqrt{x} \).

Step 1: Understand the Functions

The three functions given are: \( y=\sqrt{x} \), \( y=-\sqrt{x} \), and \( y=\sqrt{x} \sin 5 \pi x \). The first two are basic square root functions, with the second one being the reflection of the first across the x-axis. The third function is a modification of \( y=\sqrt{x} \) with a sine wave modulation.

Step 2: Graph the First Function

Graph \( y=\sqrt{x} \). This function is defined for \( x \geq 0 \) and outputs the non-negative root of \( x \). The graph is a curve that starts from the origin \((0,0)\) and rises slowly to the right.

Step 3: Graph the Second Function

Graph \( y=-\sqrt{x} \). Like \( y=\sqrt{x} \), this function is defined for \( x \geq 0 \), but outputs the negative root of \( x \). The graph starts from the origin too but goes downward as \( x \) increases.

Step 4: Graph the Third Function

Graph \( y=\sqrt{x} \sin 5 \pi x \). This function combines the square root and a sine function with a frequency of \( 5 \pi \). The graph will oscillate above and below the x-axis with an amplitude proportional to the square root of \( x \), resulting in a wave-like pattern that starts at the origin.

Step 5: Analyze the Relationships

The graph of \( y=-\sqrt{x} \) is the reflection of \( y=\sqrt{x} \) across the x-axis. The graph of \( y=\sqrt{x} \sin 5 \pi x \) is a wave-like modification of \( y=\sqrt{x} \), oscillating symmetrically around the x-axis. All three graphs originate from \((0,0)\) and are related by transformations or sinusoidal modulations.

Key Concepts

Square Root Function

Let's delve into the concept of the square root function, which is visually represented by the function \( y=\sqrt{x} \). This is a fundamental function in mathematics. It is defined only for non-negative values of \( x \), which means it will only take on real and positive outputs for \( x \geq 0 \). The graph of this function starts from the origin, the point \((0, 0)\), and gradually arcs upwards.

Key characteristics of this function include:

• Domain: \( x \geq 0 \)
• Range: \( y \geq 0 \)
• Shape: A smooth curve rising to the right.
• Symmetry: None, as it only spans across the positive x-axis.

The square root function grows slower than linear functions, as its rate of increase diminishes with larger \( x \)-values. It is especially useful in concepts involving proportional relationships and geometric scaling.

Reflection of Functions

A key transformation in graphing is the reflection of functions. In our case, the reflection is demonstrated by \( y=-\sqrt{x} \). This function is a reflection of \( y=\sqrt{x} \) across the x-axis. The result is a graph that mirrors the original square root function but is inverted.

Important points about reflection:

• The graph's domain remains the same: \( x \geq 0 \).
• The range, however, changes to \( y \leq 0 \).
• Reflection processes can change the orientation and position of a graph.
• Reflections across the x-axis involve multiplying the function by \(-1\).

This transformation is useful for exploring symmetries and relationships between positive and negative values of functions.

Sine Function Modulation

Now let's explore the sine function modulation with the example \( y=\sqrt{x} \sin 5 \pi x \). This function combines a square root and a sinusoidal function. Here, the sine function modulates the graph of the square root function, giving rise to oscillations.

Modulation characteristics in this context include:

• The \( \sin 5 \pi x \) component introduces oscillatory behavior.
• The frequency \( 5 \pi \) suggests that the sine wave cycles five times in one-unit length on the x-axis.
• Amplitude of oscillations is determined by \( \sqrt{x} \), growing as \( x \) increases.
• The graph fluctuates above and below the x-axis creating a wave pattern.

This sine modulation represents periodic variations within a larger trend, often used to model phenomena that have repetitive patterns superimposed on slow trends or progressions.

Wave Patterns in Graphs

Graphing functions like \( y=\sqrt{x} \sin 5 \pi x \) can reveal interesting wave patterns. These patterns arise due to the interplay of different mathematical components such as square roots and sine waves.

Understanding wave patterns involves recognizing:

• Frequency: Refers to how many complete cycles occur within one unit interval.
• Amplitude: Dictates the height of the peaks and troughs from the mean value.
• Phase Shift: Determines where the cycle starts in relation to the origin.
• Periodicity: Sine and cosine functions exhibit predictable repeating cycles.

These characteristics make wave patterns useful for studying both natural phenomena and engineered systems, where rhythmic cycles play a crucial role. In graphs, they can showcase both consistent oscillations and varying amplitudes based on external factors like time or other variables.