Problem 72 - Algebra 2

Question

Find the asymptotes of the graph of each equation. $$ y=\frac{1}{x}+4 $$

Step-by-Step Solution

Answer

The horizontal asymptote for the function \( y =\frac{1}{x} + 4 \) is \(y=4\), and the vertical asymptote is \(x=0\).

Step 1: Identify the Horizontal Asymptote

The horizontal asymptote of the function \(y = \frac{1}{x} + 4\) can be determined by evaluating the limit of the function as \(x\) approaches infinity. If \(x\) gets extremely large, the first part of the equation \( \frac{1}{x}\) becomes extraordinarily close to zero because any number divided by infinity equals zero . This leads to \(y \)being nearly equal to \(4\) in the limit. Therefore, the line \(y=4\) is the horizontal asymptote.

Step 2: Identify the Vertical Asymptote

For the vertical asymptotes, we want to find the values of \(x\) that will make the denominator zero since we cannot divide by zero, and the value of \(y\) will tend towards infinity. This function has \(x\) in the denominator, so we solve \(x = 0\). Hence, \(x = 0\) is the vertical asymptote for the function.

Key Concepts

Rational Functions

Rational functions are a type of function where a polynomial is divided by another polynomial. These functions often appear in algebra and can be written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not zero.

Rational functions pose unique challenges and properties, such as undefined values when the denominator equals zero, and asymptotic behavior as inputs grow large or approach the undefined values. One common example is the function \( y = \frac{1}{x} + 4 \), where \( 1/x \) is the rational part. This function demonstrates both horizontal and vertical asymptotes, which are lines that the graph approaches but never actually touches or crosses.

Understanding the concepts of asymptotes helps in analyzing these functions and predicting their behavior, even for extreme values of \(x\). For instance, identifying where the horizontal and vertical asymptotes occur can provide vital information about the end behavior of rational functions.

Horizontal Asymptotes

Horizontal asymptotes represent the value that a function approaches as the independent variable — often denoted as \(x\) — becomes very large or very small.

For the rational function in the exercise, \( y = \frac{1}{x} + 4 \), as \(x\) grows to infinity or negative infinity, the term \( \frac{1}{x} \) becomes inconsequential and approaches zero. Therefore, the function approaches the horizontal line \(y = 4\).

This occurs because, for any number divided by a tremendously large number, the result is virtually zero, leaving the constant term in the function untouched as \(x\) expands.

• Horizontal asymptotes are not boundaries; functions can cross them, especially in rational functions where asymptotic behavior is related to end behavior.
• Identifying horizontal asymptotes often involves examining the degrees of the polynomials in both the numerator and denominator.

Recognizing the horizontal asymptotes of a rational function aids in understanding its end behavior, helping to predict how the function behaves as \(x\) moves toward infinity.

Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, typically where the denominator of the rational function equals zero. They represent points where the values of \(y\) increase or decrease without bound (approach \( \infty \) or \(-\infty\)).

For the function \( y = \frac{1}{x} + 4 \), a vertical asymptote exists where \(x = 0\) because the denominator is zero, making the function undefined at this point.

• Vertical asymptotes are important indicators of discontinuity in a function.
• They divide the graph into separate, distinct regions.
• Identifying vertical asymptotes generally requires solving for values that make the denominator zero.

Vertical asymptotes are crucial for understanding the domain limitations of rational functions, as they tell us where values are forbidden, hence preventing us from evaluating the function at these points.