Problem 47 - 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 \(y=\frac{a^{3}}{x^{2}+a^{2}}\) Graph \(f(x)=\frac{a^{3}}{x^{2}+a^{2}}\) if \(a=4\)

Step-by-Step Solution

Answer

Graph \(f(x)=\frac{64}{x^{2}+16}\) with a peak at \((0,4)\) and approaching \(y=0\) as \(|x|\rightarrow ext{infinity}\).

Step 1: Substitute the given value of 'a' into the equation

The given function is \(f(x)=\frac{a^{3}}{x^{2}+a^{2}}\). We need to substitute \(a=4\) into this equation. Doing so, we get: \(f(x)=\frac{4^{3}}{x^{2}+4^{2}} = \frac{64}{x^{2}+16}\).

Step 2: Identify key features of the graph

The graph of the function \(f(x)=\frac{64}{x^{2}+16}\) has symmetry with respect to the y-axis. The horizontal asymptote occurs at \(y=0\), as the fraction approaches zero when \(x\) approaches infinity. The vertical asymptote does not exist, since the denominator \(x^{2}+16\) does not become zero. Intercept can be discerned from setting \(x=0\), yielding \(f(0)=4\).

Step 3: Sketch the graph

To graph \(f(x)=\frac{64}{x^{2}+16}\), plot the point \((0, 4)\), as this is the y-intercept. As \(x\) increases or decreases away from zero, the value of \(f(x)\) decreases, approaching the horizontal asymptote at \(y=0\). Draw the curve approaching \(y=0\) symmetrically from both the positive and negative sides of the x-axis.

Step 4: Verify the graph behavior

Verify that the graph behaves appropriately near key points: it should peak at \(x=0\) and then gradually decrease to near zero as \(x\) moves towards positive or negative infinity. Ensure the curve is symmetrical about the y-axis and consistent with the characteristics of a simple rational function.

Key Concepts

Rational Functions

Rational functions are an interesting topic in algebra. At their core, these functions are represented by the ratio of two polynomial expressions. They often take the form \( f(x) = \frac{P(x)}{Q(x)} \) where both \(P(x)\) and \(Q(x)\) are polynomials. Here, the function we are concerned with is \( f(x) = \frac{64}{x^2 + 16} \). This is a type of rational function because it is a ratio of a constant polynomial over a quadratic polynomial.

• **Polynomial in the numerator:** In this case, it's a constant value \(64\), which is actually \(4^3\).
• **Polynomial in the denominator:** This is \(x^2 + 16\), which remains positive for all real numbers."

Rational functions can exhibit a variety of behaviors, and their graphs can show interesting features such as asymptotes, which we will discuss in more detail later. What's most important about rational functions like the one we're discussing is how they behave at extreme values of \(x\) and within their domain. Because \(x^2 + 16\) never equals zero, \(f(x)\) remains defined for all real \(x\). This influences the nature of its graph and offers insights into the characteristics of the curve of Agnesi.

Graphing Techniques

Graphing a rational function involves identifying key features like intercepts, symmetry, and asymptotic behavior. The function \( f(x) = \frac{64}{x^2 + 16} \) demonstrates some wonderful graphing techniques:

• **Symmetry:** Since \(f(x)\) is an even function (meaning \(f(x) = f(-x)\)), its graph is symmetrical across the y-axis. This symmetry simplifies sketching the graph because you only need to consider one half and then mirror it.
• **Intercepts:** By setting \(x = 0\), we find \(f(0) = 4\). This is the y-intercept, and there are no x-intercepts since the function never touches the x-axis.
• **Behavior at Infinity:** As \(x\) approaches positive or negative infinity, \(f(x)\) tends to zero, which suggests the presence of a horizontal asymptote.

In graphing the function, plotting a few points can help reveal more about its shape. Aside from the intercept, evaluating \(f(x)\) at different values gives us a sense of how it curves toward the asymptotic line at \(y = 0\). Additionally, because the denominator \(x^2 + 16\) ensures positivity, the entire curve remains above the x-axis.

Asymptotes

Asymptotes are lines that a graph approaches but never truly touches. They are critical to understanding the end behavior of rational functions. In the curve of Agnesi, such lines guide the overall shape of the graph.

• **Horizontal Asymptote:** For \(f(x) = \frac{64}{x^2 + 16}\), as \(x\) approaches infinity, the \(y\)-values get closer and closer to zero. This forms a horizontal asymptote at \(y = 0\).
• **Vertical Asymptote:** Vertical asymptotes occur where the denominator of a rational function equals zero. However, since \(x^2 + 16\) is always positive, there are no vertical asymptotes here.

Understanding these asymptotic behaviors is key in sketching and analyzing the function graphically. While the horizontal asymptote tells us what happens at extreme \(x\)-values, the absence of vertical asymptotes provides insights into how the function behaves over its entire domain. For instance, it assures us that the curve never diverges to infinity at any point, making the graph of \(f(x)\) smooth and continuous across all real \(x\). Such features make the curve of Agnesi both fascinating and unique.