Problem 44 - 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. $$ f(x)=\frac{3}{(x+1)^{2}} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = -1\); horizontal asymptote at \(y = 0\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero. For the function \( f(x) = \frac{3}{(x+1)^2} \), set the denominator equal to zero: \((x+1)^2 = 0\), which implies \(x+1 = 0\). Solving this gives \(x = -1\). Therefore, the vertical asymptote is at \(x = -1\).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes are found based on the degrees of the polynomials in the numerator and denominator. In \( f(x) = \frac{3}{(x+1)^2} \), the degree of the numerator is 0 (since it's a constant) and the degree of the denominator is 2. Since the degree of the denominator (2) is greater than the degree of the numerator (0), the horizontal asymptote is \(y = 0\).

Step 3: Sketch the Graph

To sketch the graph, start by plotting the vertical asymptote at \(x = -1\) and the horizontal asymptote at \(y = 0\). Recognize that \(f(x)\) approaches \(0\) (the horizontal asymptote) as \(x\) tends toward positive or negative infinity. The graph will be shaped like an upside-down U on both sides of the vertical asymptote, descending toward the horizontal asymptote as \(x\) moves away from \(-1\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are crucial in understanding the behavior of rational functions. They occur at specific values of the input, where the function does not exist and tends towards infinity. This happens when the denominator of a rational function equals zero, while the numerator remains non-zero. In the function \( f(x) = \frac{3}{(x+1)^2} \), the denominator is \((x+1)^2\). Zeroing the denominator gives the equation \((x+1)^2 = 0\), leading to \(x = -1\).
At \(x = -1\), the function spikes upwards or downwards towards infinity, creating a vertical asymptote. This vertical line marks a boundary the graph never crosses.
You can easily identify vertical asymptotes through these steps:

• Set the denominator equal to zero.
• Solve for \(x\).
• Each solution is a vertical asymptote when the numerator is non-zero at those points.

These asymptotes are useful for sketching the general shape and behavior of the graph.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as the input \(x\) moves towards positive or negative infinity. In simpler terms, they tell us what value the function's output is approaching.
To find horizontal asymptotes, we compare the degrees of the polynomials in the numerator and denominator. In the function \( f(x) = \frac{3}{(x+1)^2} \), the numerator is a constant \(3\) and has a degree of 0. The denominator \((x+1)^2\) has a degree of 2.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there's no horizontal asymptote, but there might be an oblique one instead.

For our function here, since the denominator’s degree is greater, the horizontal asymptote is \(y = 0\). This indicates as \(x\) approaches infinity in either direction, the output approaches zero.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. The behavior of these functions, particularly asymptotes, gives significant insight into their graphing.
Analyzing the rational function \( f(x) = \frac{3}{(x+1)^2} \) highlights two important parts: the numerator selected as 3, and the denominator \((x+1)^2\).
To understand any rational function:

• Look at the zeros of the denominator to find vertical asymptotes.
• Compare the degrees of the numerator and denominator to identify horizontal asymptotes.
• Consider any values that might cancel out, known as removable discontinuities, though they do not apply to this example.

Rational functions often form curves that approach but do not cross asymptotes, creating interesting bends and turns in a graph.
Understanding these functions helps predict the direction and ultimate behavior of their graphs as \(x\) values become extremely large or small.