Problem 48 - Algebra 2

Question

HISTORY For Exercises \(47-49\) , use the following information. In Maria Gaetana Agnesi's book Analytical Institutions, Agnesi discussed the characteristics of the equation \(x^{2} y=a^{2}(a-y),\) the graph of which is called the "curve of Agnesi." This equation can be expressed as \(y=\frac{a^{3}}{x^{2}+a^{2}}\) Describe the graph. What are the limitations on the domain and range?

Step-by-Step Solution

Answer

The domain is all real numbers; the range is \(0 < y \leq a\). The graph is symmetric about the y-axis.

Step 1: Understand the Equation

The equation for the "curve of Agnesi" is given as \( y = \frac{a^3}{x^2 + a^2} \). This is a rational function where \(a\) is a constant.

Step 2: Analyze the Domain

The domain of a rational function is determined by the values of \(x\) that do not make the denominator zero. Here, the denominator is \(x^2 + a^2\). Since \(x^2 + a^2 eq 0\) for all real \(x\) (assuming \(a eq 0\)), the domain of the function is all real numbers \(x \in (-\infty, \infty)\).

Step 3: Determine the Range

To find the range, consider the behavior of \(y\). As \(x\) approaches 0, \(y\) approaches \(\frac{a^3}{a^2} = a\). As \(|x|\) increases, \(y\) approaches 0. Thus, the range is \(0 < y \leq a\).

Step 4: Describe the Graph

The graph of the function \(y = \frac{a^3}{x^2 + a^2}\) is a symmetric curve about the y-axis because \(x^2\) is symmetric. It approaches \(y = 0\) as \(|x|\) becomes very large and reaches a maximum value of \(a\) at \(x = 0\).

Key Concepts

Rational Functions

A rational function is essentially a fraction where the numerator and the denominator are both polynomials. These functions can have interesting characteristics based on their structure and the interaction of their polynomial components. In our study of the "curve of Agnesi," the function is given by the equation:

• \( y = \frac{a^3}{x^2 + a^2} \)

Here, the numerator is a constant \( a^3 \) and the denominator is \( x^2 + a^2 \). This setup implies that the function behaves like a typical rational function, with potential constraints based on the values that the denominator can take.
Rational functions can exhibit vertical asymptotes, where the function is undefined, and horizontal or oblique asymptotes, showing the end behavior of the graph. They also often have sections where they increase or decrease, and domains and ranges specific to their configurations. Understanding these will help in graphing and interpreting such functions.

Domain and Range

The domain and range are essential elements of the function's analysis. The **domain** of a function refers to all the possible input values (usually \( x \)) that the function can handle without running into mathematical issues, such as division by zero.
In the "curve of Agnesi," the function defined is:

• \( y = \frac{a^3}{x^2 + a^2} \)

The denominator \( x^2 + a^2 \) cannot be zero, but since \( x^2 + a^2 \) will never be zero for real numbers (unless \( a \) were zero or undefined), the domain is all real numbers, \( x \in (-\infty, \infty) \).
The **range** refers to all possible output values (\( y \)) a function can produce. Here, as \( x \) approaches zero, \( y \) reaches its maximum value, \( a \). As \( |x| \) increases, the value of \( y \) approaches zero. Thus, the range of the function is restricted to \( 0 < y \leq a \). Understanding the range and domain helps in visualizing and predicting the behavior of a graph.

Symmetry in Graphs

Symmetry is a crucial aspect that can simplify the analysis and understanding of a graph. The graph of the "curve of Agnesi" showcases a specific type of symmetry known as **even symmetry**. This occurs when graphs are mirrored around the y-axis.
In our equation:

• \( y = \frac{a^3}{x^2 + a^2} \)

The presence of \( x^2 \) in the denominator indicates that plugging in \( x \) or \(-x\) yields the same value for \( y \). Hence, it shows symmetry about the y-axis.
Identifying symmetry helps in sketching graphs since calculations for one side of the graph can inform you about the other side automatically. Analyzing symmetry is an efficient way to predict and confirm the behavior of functions over their respective domains.