Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ F(x)=\frac{3}{9-x^{2}} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = \pm 3\); Horizontal asymptote: \(y = 0\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, typically where the denominator equals zero. For the function \( F(x) = \frac{3}{9 - x^{2}} \), set the denominator equal to zero: \[9 - x^2 = 0\] Solving for \(x\), we get: \[9 = x^2\] \[x = \pm 3\] So, the vertical asymptotes are at \(x = 3\) and \(x = -3\).

Step 2: Identify Horizontal Asymptotes

Horizontal asymptotes are found by examining the behavior of \(F(x)\) as \(x\) goes to \(+\infty\) or \(-\infty\). Since the degree of the polynomial in the denominator (2) is greater than that in the numerator (0), the horizontal asymptote is at \(y = 0\) (the x-axis).

Step 3: Sketch the Graph

To sketch the graph, draw the vertical asymptotes as dashed lines at \(x = 3\) and \(x = -3\), and the horizontal asymptote as a dashed line along the x-axis at \(y = 0\). The function \( F(x) = \frac{3}{9 - x^{2}} \) is undefined at \(x = \pm 3\) and approaches \(0\) as \(x\) goes to \(\pm\infty\). Draw the curve approaching these asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are essential in understanding the boundaries of a function's behavior. They occur at the values of \( x \) where the function's denominator equals zero, leading to the function being undefined. For the function \( F(x) = \frac{3}{9 - x^2} \), you find the vertical asymptotes by setting the denominator \( 9 - x^2 \) equal to zero. Solving the equation \( 9 = x^2 \) gives \( x = \pm 3 \). This means the graph of the function has vertical asymptotes at \( x = 3 \) and \( x = -3 \).

• Vertical asymptotes occur when the denominator is zero.
• For \( F(x) = \frac{3}{9-x^2} \), vertical asymptotes are at \( x = 3 \) and \( x = -3 \).
• The function is undefined at these points.

These asymptotes divide the graph into distinct sections, where you typically see the function approaching either positive or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a function as \( x \) approaches \( +\infty \) or \( -\infty \). They tell us what value the function is heading towards on the y-axis. To find a horizontal asymptote for rational functions, we primarily compare the degrees of the polynomials in the numerator and the denominator.In the function \( F(x) = \frac{3}{9-x^2} \), the numerator has a degree of 0, and the denominator has a degree of 2. When the degree in the denominator is greater than that in the numerator, the horizontal asymptote will always be at \( y = 0 \) or the x-axis.

• Horizontal asymptotes show end behavior of the graph.
• Found by comparing the degrees of numerator and denominator.
• For this function, \( y = 0 \) is the horizontal asymptote.

This means, as \( x \) becomes very large or very small, the graph levels off and becomes closer and closer to the x-axis, never actually touching it.

Function Sketching

When sketching a rational function, the identification of asymptotes helps structure the graph efficiently. Start by drawing the asymptotes you've found:

• Vertical asymptotes as dashed lines at \( x = 3 \) and \( x = -3 \).
• A horizontal asymptote as a dashed line at \( y = 0 \).

Now, consider how the function behaves around these asymptotes. For \( F(x) = \frac{3}{9 - x^2} \), the function is undefined at the asymptote locations, meaning it will tend toward infinity as it approaches \( x = 3 \) and \( x = -3 \).The function approaches 0 as \( x \) moves away from central values towards \( +\infty \) or \( -\infty \).

• Draw the curve nearing vertical asymptotes but never crossing them.
• Let the curve level close to the horizontal asymptote \( y = 0 \) as \( x \) tends to \( \pm\infty \).

This approach helps to conceptualize how the graph behaves, giving a clearer visualization.

Rational Functions

Rational functions are ratios of two polynomials and have specific properties that make their behavior predictable and fascinating. These functions often have asymptotes, critical points where the graph may turn or slope quickly as it approaches infinity.

• Written as \( F(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials.
• Vertical asymptotes occur where \( q(x) = 0 \).
• Horizontal asymptotes depend on the degrees of \( p(x) \) and \( q(x) \).

In the case of \( F(x) = \frac{3}{9 - x^2} \), the structure reveals it has two vertical asymptotes due to the quadratic denominator. The lack of a higher degree in the numerator means the horizontal asymptote is the x-axis.Understanding rational functions helps predict how they should visually appear on a graph, their limits, and their potential to rise or fall sharply. This makes analyzing real-world applications easier as they display similar characteristics in physical systems and natural processes.