Problem 49 - 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 \(y=\frac{a^{3}}{x^{2}+a^{2}}\) Make a conjecture about the shape of the graph of \(f(x)=\frac{a^{3}}{x^{2}+a^{2}} \mathrm{ff} a=-4\) Explain your reasoning.

Step-by-Step Solution

Answer

The graph is an inverted bell curve, symmetrical about the y-axis.

Step 1: Substitute the value of 'a'

The equation given is \( f(x) = \frac{a^{3}}{x^{2} + a^{2}} \). We need to substitute \( a = -4 \) into this equation. So, \(-4^3 = -64\) and \((-4)^2 = 16\), leading to the equation \( f(x) = \frac{-64}{x^{2} + 16} \).

Step 2: Analyze the Denominator

For the function \( f(x) = \frac{-64}{x^2 + 16} \), note that the denominator \( x^2 + 16 \) is always positive for all real numbers, since a square term cannot be negative and 16 is a positive number.

Step 3: Identify the Asymptotes and Domain

This function does not have vertical asymptotes because the denominator \( x^2 + 16 \) never equals zero. The domain of the function is all real numbers \( x \in \mathbb{R} \).

Step 4: Determine the Shape of the Graph

As \(x\) approaches positive or negative infinity, \( f(x) \) approaches zero because the denominator grows much faster than the numerator. This suggests a horizontal asymptote of \( y = 0 \).

Step 5: Plot Critical Points

Calculate \( f(x) \) at specific points to understand the curve's shape. For \( x = 0 \), \( f(0) = \frac{-64}{16} = -4 \). Symmetry can be observed, as \( f(x) = f(-x) \), indicating the function is even.

Step 6: Make the Conjecture

Based on the above analysis, the graph of the function \( f(x) = \frac{-64}{x^2 + 16} \) will be similar to an inverted bell curve (like a Gaussian curve) symmetrical about the y-axis, approaching \( y = 0 \) as \( x \to \infty \) or \( x \to -\infty \).

Key Concepts

Asymptotes

Understanding asymptotes is crucial when analyzing function graphs. Asymptotes are lines that a graph approaches but never actually touches. They can be either vertical, horizontal, or oblique (slant). In the function \( f(x) = \frac{-64}{x^2 + 16} \), we need to identify whether there are asymptotes and their type.
- **Vertical Asymptotes**: These occur when the denominator equals zero, causing the function to become undefined. However, for our equation, \( x^2 + 16 \) is never zero because any real number squared is non-negative and adding 16 keeps it positive. Hence, there are no vertical asymptotes.
- **Horizontal Asymptotes**: These are determined by the behavior of the function as \( x \) approaches positive or negative infinity. Since the degree of the polynomial in the denominator (2) is larger than the numerator (0), \( f(x) \) approaches zero as \( x \to \pm \infty \). Therefore, there is a horizontal asymptote at \( y = 0 \).
Identifying these lines can help determine the overall shape and behavior of the graph.

Domain and Range

The domain of a function represents all the possible input values \( x \) for which the function is defined. For \( f(x) = \frac{-64}{x^2 + 16} \), we have seen that the denominator is always positive. There are no restrictions on the values that \( x \) can take, meaning the domain is all real numbers: \( x \in \mathbb{R} \).
The range is the set of possible output values \( y \). To find out the range, we consider the behavior of the function over its domain. As the denominator \( x^2 + 16 \) increases for any \( x \), the whole fraction decreases in magnitude since \(-64\) is a constant numerator. The largest value the function attains is \(-4\) when \( x = 0 \), because \( f(0) = -4 \). For any other \( x \), \( f(x) \) is less than \(-4\), approaching zero.
Thus, the range of the function can be defined as \( (-\infty, -4] \). This reflects how tightly the graph is bound below by the curve's behavior.

Symmetry in Graphs

Symmetry in graphs helps simplify the understanding of graph behaviors and characteristics. A function is termed symmetric concerning the y-axis if \( f(x) = f(-x) \) for all x in its domain. This type of symmetry indicates that the function is even.
For the given function \( f(x) = \frac{-64}{x^2 + 16} \), we can test for symmetry by substituting \(-x\) in place of \(x\). The function becomes \( \frac{-64}{(-x)^2 + 16} \), which simplifies back to the original function \( \frac{-64}{x^2 + 16} \). Since \( f(x) = f(-x) \), the graph is symmetric about the y-axis.
Understanding that the curve has y-axis symmetry can aid in predicting its shape. For instance, when plotting points for the graph, it’s sufficient to compute values for positive \( x \), since the negative \( x \) values will mirror them. This insight into symmetry makes graph sketching an easier endeavor.