Problem 24 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{3 x^{2}}{x^{2}-9} $$

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = 3 \) and \( x = -3 \); horizontal asymptote at \( y = 3 \).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function is equal to zero, provided that the numerator is not zero at these points. For the function \( f(x)=\frac{3x^2}{x^2-9} \), set the denominator equal to zero: \( x^2 - 9 = 0 \). Solving gives \( x^2 = 9 \), so \( x = 3 \) and \( x = -3 \). Therefore, there are vertical asymptotes at \( x = 3 \) and \( x = -3 \).

Step 2: Check for Horizontal Asymptotes

Horizontal asymptotes depend on the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) is the leading coefficient of the numerator and \( b \) is the leading coefficient of the denominator. Here, both polynomials are of degree 2, and the leading coefficients are both 3 and 1, respectively. Thus, there is a horizontal asymptote at \( y = \frac{3}{1} = 3 \).

Key Concepts

Rational Functions

Rational functions are a fundamental type of mathematical function defined as the quotient of two polynomials. In simpler terms, it is a fraction where both the numerator and the denominator are polynomials. For example, in the function \( f(x) = \frac{3x^2}{x^2 - 9} \), \( 3x^2 \) is the numerator and \( x^2 - 9 \) is the denominator.

• The numerator determines the zeros of the function when it equals zero, which are the x-values where the function itself equals zero.
• The denominator dictates the x-values that make the function undefined, which are crucial for identifying asymptotes.

Rational functions can exhibit behavior such as vertical or horizontal asymptotes, depending on the relationship between their numerator and denominator. Understanding these properties is key to analyzing and graphing rational functions.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, making the function undefined. Unlike other vertical lines, these are not part of the graph but indicate where the function rises or falls indefinitely. For \( f(x) = \frac{3x^2}{x^2 - 9} \), setting the denominator \( x^2 - 9 \) equal to zero gives the equation \( x^2 = 9 \).

• Solving for \( x \), we find \( x = 3 \) and \( x = -3 \), which means there are vertical asymptotes at these x-values.
• At these points, the function "blows up," meaning it approaches positive or negative infinity.

It's crucial to remember vertical asymptotes do not intersect the graph of the function but highlight approaches towards undefined behavior. These lines significantly affect the function's graph, restricting its continuity.

Horizontal Asymptotes

Horizontal asymptotes tell us how a rational function behaves as the input \( x \) approaches extremely large or small values. They are not barriers like vertical asymptotes but rather guides to the end behavior of the function. For example, in \( f(x) = \frac{3x^2}{x^2 - 9} \), both numerator and denominator are polynomials of degree 2, with leading coefficients 3 and 1, respectively.

• When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by \( y = \frac{3}{1} = 3 \).
• This means as \( x \) becomes very large or small, \( f(x) \) approaches, but never actually reaches, the line \( y = 3 \).

Horizontal asymptotes are particularly useful for predicting the long-term direction of the function's graph and understanding its limits as \( x \) approaches positive or negative infinity.