Problem 20 - APEX Calculus

Question

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{-3 x^{2}-9 x-6}{5 x^{2}-10 x-15} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the form of the rational function

The given function is a rational function of the form \( f(x) = \frac{-3x^2 - 9x - 6}{5x^2 - 10x - 15} \). Rational functions can exhibit vertical or horizontal asymptotes depending on the degree of the polynomial in the numerator relative to the denominator.

Step 2: Determine the vertical asymptotes

Vertical asymptotes occur when the denominator is zero and the numerator is non-zero. To find vertical asymptotes, we solve the equation \( 5x^2 - 10x - 15 = 0 \). Factor the denominator: \( 5(x^2 - 2x - 3) = 5(x-3)(x+1) \). Setting each factor to zero gives \( x = 3 \) and \( x = -1 \) as the points where vertical asymptotes occur.

Step 3: Determine the horizontal asymptotes

Horizontal asymptotes depend on the degrees of the polynomials in the numerator and denominator. If their degrees are the same, then the horizontal asymptote is \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients. In this function, both the numerator and the denominator have degree 2. The leading coefficients are \(-3\) and \(5\), respectively. Hence, the horizontal asymptote is \( y = \frac{-3}{5} \).

Key Concepts

Vertical Asymptotes

When we talk about vertical asymptotes in a rational function like \( f(x) = \frac{-3x^2 - 9x - 6}{5x^2 - 10x - 15} \), we are looking for values of \( x \) where the function is undefined. These occur at the zeros of the denominator.

To find vertical asymptotes, set the denominator equal to zero and solve for \( x \):
\[ 5x^2 - 10x - 15 = 0 \]
Factor the quadratic expression:

• Start by factoring out the greatest common factor: \( 5(x^2 - 2x - 3) \).
  
• Further factor the expression: \( (x - 3)(x + 1) \).
  

By setting each factor equal to zero, we find:

• \( x - 3 = 0 \) leads to \( x = 3 \)
  
• \( x + 1 = 0 \) leads to \( x = -1 \)
  

Thus, vertical asymptotes are at \( x = 3 \) and \( x = -1 \).

These asymptotes represent values where the graph of the function will sharply approach but never actually touch, showing infinite behavior at these points.

Horizontal Asymptotes

Horizontal asymptotes indicate the behavior of a function as \( x \) approaches positive or negative infinity. For rational functions, the degree of the numerator and the degree of the denominator play a critical role.

In the function \( f(x) = \frac{-3x^2 - 9x - 6}{5x^2 - 10x - 15} \), the degree of both the numerator and denominator is 2. This means we have a specific rule:

• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively.
  

Here, the leading coefficient in the numerator is \(-3\) and in the denominator is \(5\). Therefore, the horizontal asymptote is given by:
\[ y = \frac{-3}{5} \]

This tells us that as \( x \) progresses towards infinity in either direction, the value of \( f(x) \) will approach \( \frac{-3}{5} \), giving us a y-value the graph will tend to flatten towards but not cross.

Rational Functions

Rational functions are a key concept in algebra, defined as the ratio of two polynomials. In general, a rational function has the form:
\[ f(x) = \frac{P(x)}{Q(x)} \]
where \( P(x) \) and \( Q(x) \) are polynomials. The domain of rational functions excludes any values of \( x \) that make \( Q(x) = 0 \), as these lead to undefined expressions.

Rational functions can exhibit different types of asymptotes:

• Vertical asymptotes happen where the denominator is zero but the numerator is not, resulting in undefined behavior.
  
• Horizontal or slant asymptotes describe the end behavior of the function as \( x \) approaches infinity.
  

These features are key to sketching the graph of the function. Understanding this helps students predict how the function behaves, aligns approaching lines, and highlights areas where the function won't return a defined value.