Problem 64 - Algebra and Trigonometry

Question

\(61-66\). Graph the three functions on a common screen. How are the graphs related? $$ y=\frac{1}{1+x^{2}}, \quad y=-\frac{1}{1+x^{2}}, \quad y=\frac{\cos 2 \pi x}{1+x^{2}} $$

Step-by-Step Solution

Answer

The first two graphs are reflections across the x-axis. The third graph oscillates and decreases in amplitude like the first.

Step 1 – Understand the Functions

First, let's identify the functions we'll be working with:1. \(y = \frac{1}{1+x^2}\)2. \(y = -\frac{1}{1+x^2}\)3. \(y = \frac{\cos(2\pi x)}{1+x^2}\).Each function involves the term \(\frac{1}{1+x^2}\), which is similar to the form of a standard rational function.

Step 2 – Graph the First Function

Graph the function \(y = \frac{1}{1+x^2}\). This function is symmetric about the y-axis, reaching a maximum value of 1 at \(x = 0\) and decreasing as \(x\) moves away from zero. It forms a bell-shaped curve, approaching zero as \(|x|\) becomes large.

Step 3 – Graph the Second Function

For the graph of \(y = -\frac{1}{1+x^2}\), take the negative of the first function. This graph is simply the reflection of \(y = \frac{1}{1+x^2}\) across the x-axis. It has a minimum value of -1 when \(x = 0\) and approaches zero from below as \(|x|\) increases.

Step 4 – Graph the Third Function

Graph \(y = \frac{\cos(2\pi x)}{1+x^2}\). This graph combines the oscillation caused by \(\cos(2\pi x)\) with the diminishing factor \(\frac{1}{1+x^2}\). The oscillations decrease in amplitude as \(x\) increases, similar to an exponentially decaying sine wave modulated by a rational function.

Step 5 – Analyze Graph Relationships

Compare the graphs on a common screen: - The first and second graphs are reflections of each other across the x-axis. - The third graph has similar decaying characteristics to the first two but oscillates between positive and negative values. - This oscillation is due to the cosine function which modulates the amplitude of the decay.

Key Concepts

Symmetric Functions

Symmetry is a key concept in mathematics, especially when analyzing graphs. A function is symmetric if it can be mapped onto itself through an operation such as reflection. For the exercise at hand, the first function, \(y = \frac{1}{1+x^2}\), is symmetric about the y-axis. This means that its graph looks the same on both sides of the y-axis; this is called even symmetry. You can test for symmetry about the y-axis by checking if \(f(x) = f(-x)\) for all x in the function's domain.
In contrast, the graph of the second function, \(y = -\frac{1}{1+x^2}\), is symmetric with respect to the origin. This happens because it's essentially the same graph as the first one but flipped over the x-axis. This demonstrates what's known as odd symmetry, where \(f(x) = -f(-x)\) holds true. Comparing these symmetries helps in predicting the behavior and shape of functions without needing complex calculations.

Rational Functions

Rational functions are quotients of polynomials, such as \(y = \frac{1}{1+x^2}\). They often have interesting features, including asymptotic behavior and unique shapes. Here, the basic rational function \(y = \frac{1}{1+x^2}\) forms a bell-shaped curve. This particular curve does not have vertical asymptotes because the denominator never reaches zero. Instead, it approaches zero as \(|x|\) becomes very large.
When a rational function includes additional terms, such as trigonometric factors, the complexity increases but the basic behavior remains similar. For example, the function \(y = \frac{\cos 2\pi x}{1+x^2}\) shares its denominator with the simpler functions, ensuring a similar decay pattern. Understanding this core structure in rational functions allows us to predict where the function reaches its extremum and how it behaves at large \(|x|\).

Trigonometric Functions

Trigonometric functions, like cosine, add complexity to rational functions through oscillation. The third function \(y = \frac{\cos 2\pi x}{1+x^2}\) demonstrates how this oscillatory behavior is modulated. The \(\cos(2\pi x)\) part ensures the graph will oscillate between positive and negative values, bringing about periodic peaks and troughs, typical of cosine functions.
The periodic nature of the cosine function, with a period of 1 due to the \(2\pi x\) inside the cosine, adds a dynamic wave-like feature over the diminishing rational base function. As \(|x|\) becomes large, the influence of the cosine part is less pronounced, as the overall function's amplitude shrinks towards zero, causing the oscillations to fade. This combination is akin to a transient damped wave, displaying both periodicity and asymptotic decay.