Problem 30 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). $$ r(x)=\frac{5 x^{3}}{x^{3}+2 x^{2}+5 x} $$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 0 \); horizontal asymptote at \( y = 5 \).

Step 1: Simplify the Rational Function

First, we simplify the expression if possible. Looking at \( r(x)=\frac{5x^3}{x^3+2x^2+5x} \), we can factor the denominator as \( x(x^2 + 2x + 5) \). There is no common factor in the numerator and the denominator, so the function is already in its simplest form.

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero but the numerator is not. Solve \( x^3+2x^2+5x=0 \) for \( x \):\[ x(x^2 + 2x + 5) = 0 \]This equation gives \( x = 0 \). The quadratic \( x^2 + 2x + 5 = 0 \) has solutions given by the quadratic formula, but its discriminant \( 2^2 - 4 \cdot 1 \cdot 5 = -16 \) is negative, implying no real roots. Therefore, there is only one vertical asymptote at \( x = 0 \).

Step 3: Find Horizontal Asymptotes

Horizontal asymptotes are determined by the degrees of the polynomials in the numerator and denominator. Since both numerators and denominators have the same degree (3), divide the leading coefficients: \( \frac{5}{1} = 5 \). This means there is a horizontal asymptote at \( y = 5 \).

Step 4: Determination of End Behavior

Since the degrees of the numerator and denominator are the same, the horizontal asymptote represents the end behavior of the rational function. Hence, as \( x \to \infty \) or \( x \to -\infty \), \( r(x) \to 5 \).

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. Understanding these functions is crucial because they often model real-world scenarios. In the function \( r(x) = \frac{5x^3}{x^3 + 2x^2 + 5x} \), the numerator has the polynomial \(5x^3\), and the denominator is \(x^3 + 2x^2 + 5x\).
These functions can exhibit interesting properties, such as asymptotes, which are lines that the graph of the function approaches but never actually reaches.
When dealing with rational functions, it's important to first simplify the expression, if possible. Simplifying might involve factoring both numerator and denominator and canceling out any common terms. In our function, there are no common factors to cancel, so it remains as is.
Rational functions are particularly noteworthy for their ability to produce curves with segments going off to infinity, which creates the need to understand how they behave across their domains, particularly around points where asymptotes occur.

Vertical Asymptotes

Vertical asymptotes are the values of \(x\) at which a rational function becomes undefined, typically when the denominator equals zero, leading to a division by zero. For the function \( r(x) = \frac{5x^3}{x^3 + 2x^2 + 5x} \), we investigate where \(x^3 + 2x^2 + 5x = 0\) because these values will reveal our vertical asymptotes.

First, factoring the denominator gives \(x(x^2 + 2x + 5) = 0\). Solving \(x = 0\) indicates that there is a vertical asymptote on the graph at \(x = 0\).
The quadratic \(x^2 + 2x + 5\) is solved using the quadratic formula, but its negative discriminant \(-16\) reveals no real roots, confirming \(x = 0\) as the only vertical asymptote.
It's key to note that a function graph approaches but never crosses a vertical asymptote. This behavior is due to the graph moving vertically toward infinity or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes describe how a rational function behaves as \(x\) approaches infinity or negative infinity. For the function \( r(x) = \frac{5x^3}{x^3 + 2x^2 + 5x} \), we note that the degrees of the numerator and the denominator are the same (both are 3).

When the degrees are equal, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and the denominator. In this case, it's \(\frac{5}{1} = 5\), indicating a horizontal asymptote at \(y = 5\).
As \(x\) heads to infinity or negative infinity, \(r(x)\) will approach 5, defining the end behavior of this rational function.
Horizontal asymptotes provide a great way to visualize how the function behaves at extreme values of \(x\), giving us a clear picture of the overall shape and direction of the graph at its tail ends.