Problem 27 - Precalculus Mathematics for Calculus

Question

Find all horizontal and vertical asymptotes (if any). $$s(x)=\frac{(5 x-1)(x+1)}{(3 x-1)(x+2)}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = \frac{1}{3} \) and \( x = -2 \); horizontal asymptote at \( y = \frac{5}{3} \).

Step 1: Determine Vertical Asymptotes

Vertical asymptotes occur when the denominator is equal to zero. Solve the equation \( (3x-1)(x+2) = 0 \) to find the values of \( x \). This occurs when:\[3x-1=0 \quad \Rightarrow \quad x=\frac{1}{3}\]\[x+2=0 \quad \Rightarrow \quad x=-2\]Thus, the function has vertical asymptotes at \( x = \frac{1}{3} \) and \( x = -2 \).

Step 2: Check for Horizontal Asymptotes

Horizontal asymptotes depend on the degrees of the polynomials in the numerator and denominator. The degree of the numerator is 2 and the degree of the denominator is 2:\[ s(x) = \frac{5x^2 + 4x - 1}{3x^2 + 5x - 2}\]Since both degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Thus, the horizontal asymptote is:\[y = \frac{5}{3}\]

Step 3: Conclusion on Asymptotes

The function \( s(x) \) has vertical asymptotes at \( x = \frac{1}{3} \) and \( x = -2 \), and a horizontal asymptote at \( y = \frac{5}{3} \).

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes in rational functions become a point of analysis when the output of the function approaches a particular value as the input becomes very large. This behavior is directly related to the degrees of the polynomials found in the numerator and the denominator of the rational function.
To find horizontal asymptotes, compare the degree of the polynomial in the numerator to the degree of the polynomial in the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at the line \( y = 0 \).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of these polynomials.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In the example given, the function \( s(x) = \frac{(5x-1)(x+1)}{(3x-1)(x+2)} \) simplifies to \( \frac{5x^2 + 4x - 1}{3x^2 + 5x - 2} \). Here, both polynomials have a degree of 2, which means that the horizontal asymptote is at \( y = \frac{5}{3} \), the ratio of the leading coefficients \( 5 \) and \( 3 \). This provides us a horizontal line that the graph of the function approaches but never crosses as \( x \) tends to infinity or negative infinity.

Vertical Asymptotes

Vertical asymptotes occur in rational functions when the denominator equals zero, making the function undefined at particular points. These asymptotes represent lines that the function values sharply rise or fall towards without actually reaching or crossing these lines.
To determine vertical asymptotes, set the denominator of the rational function to zero and solve for \( x \). In the function \( s(x) = \frac{(5x-1)(x+1)}{(3x-1)(x+2)} \), we set \((3x-1)(x+2) = 0\) to find the vertical asymptotes.

• Solving \( 3x-1 = 0 \) gives \( x = \frac{1}{3} \)
• Solving \( x+2 = 0 \) gives \( x = -2 \)

Therefore, the vertical asymptotes are at \( x = \frac{1}{3} \) and \( x = -2 \). These values cause the denominator to become zero, leading to undefined points and thus, vertical asymptotes in the graph. The function will approach but not cross these vertical lines, creating a distinct division in the behavior of the function.

Polynomial Degrees

Understanding polynomial degrees is crucial when dealing with rational functions, particularly in finding asymptotes. The degree of a polynomial is the highest power of the variable in the polynomial equation. It provides a clear understanding of the polynomial's growth and its dominant term.
When examining rational functions, knowing the degrees of both the numerator and denominator polynomials can offer insightful information about the function's long-term behavior:

• The degree helps determine the existence and location of horizontal asymptotes. If degrees are equal, focus on the leading coefficients.
• It helps understand the overall shape of the graph and guides the prediction of end behavior.
• The interaction of these degrees influences whether there are horizontal or slant asymptotes, if applicable.

In our rational function \( s(x) = \frac{(5x-1)(x+1)}{(3x-1)(x+2)} \), both the numerator and denominator simplify to quadratic polynomials with a degree of 2. By analyzing these degrees, it's evident that this function will have a horizontal asymptote at the ratio of the leading coefficient, \( y = \frac{5}{3} \). The degree gives a structured approach to foresee the behavior of rational functions as \( x \) moves towards positive or negative infinity.