Problem 128 - Algebra and Trigonometry

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

Step-by-Step Solution

Answer

The statement 'The graph of a rational function can have three vertical asymptotes.' is true.

Step 1: Understanding Rational Functions and Vertical Asymptotes

A rational function is defined as \( f(x) = p(x)/q(x) \) where \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) \) is not equal to zero. A vertical asymptote refers to the values of \( x \) at which the function goes to infinity, that is, it occurs at the zeroes of the denominator \( q(x) \).

Step 2: Evaluating the statement

Given the nature of rational functions, there is no implicit limit to the number of vertical asymptotes that a graph can have. This means that the graph of a rational function can indeed have three vertical asymptotes. The number of vertical asymptotes depends on the degree of the denominator, that is, the number of distinct roots it has.

Step 3: Formulating the Result

Since it is possible for a rational function to have three vertical asymptotes, the statement is true and requires no changes to make it true.

Key Concepts

Understanding Vertical Asymptotes in Rational Functions

A vertical asymptote represents a value of \( x \) where the function approaches infinity or negative infinity. This is a key feature of rational functions. To understand vertical asymptotes in the context of rational functions, consider the function \( f(x) = \frac{p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomials. The vertical asymptotes occur where the denominator \( q(x) \) is equal to zero but \( p(x) \) is not. These are the points where the function diverges.

• Vertical asymptotes are graphical representations where the function values shoot up to extreme levels (positive or negative) at certain \( x \)-values.
• Vertical asymptotes occur at the roots of the denominator \( q(x) \), assuming they are not canceled out by the numerator \( p(x) \).
• The number of vertical asymptotes is determined by the number of distinct roots in the denominator polynomial.

In conclusion, a rational function can have one, two, or even more vertical asymptotes. The important thing is that these asymptotes are determined by the behavior of the denominator polynomial.

Explaining Polynomials in Rational Functions

Polynomials are essential in forming rational functions. They are expressions made from variables and coefficients, combined using operations like addition, subtraction, and multiplication. A polynomial can be represented as \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \) for the numerator, and similarly for the denominator \( q(x) \).

• Polynomials are characterized by their degree, which is the highest power of \( x \) in the expression.
• A higher degree in the denominator polynomial means more potential vertical asymptotes.
• The roots of the polynomial \( q(x) \) define where vertical asymptotes of the function may occur.

Understanding polynomials helps in predicting the behavior of the rational function's graph, especially in identifying vertical asymptotes and other features.

Finding Denominator Roots and Vertical Asymptotes

To identify vertical asymptotes of a rational function, focus on finding the roots of the denominator polynomial \( q(x) \). These roots are critical as they signal potential spots for asymptotes. Here's how to do it:

• Set the denominator polynomial equal to zero: \( q(x) = 0 \).
• Solve this equation to find the values of \( x \) where the function is undefined. These are the denominator roots.
• Check that the specific root is not canceled out by a root from the numerator \( p(x) \). If it is not canceled, it will be a vertical asymptote.

For example, if \( q(x) = x^2 - 9 \), then the roots are found by solving the equation \( x^2 - 9 = 0 \), resulting in \( x = 3 \) and \( x = -3 \). Both values represent vertical asymptotes, provided they are not canceled by the numerator. Identifying these roots correctly is key to understanding the graph's behavior and where it will show vertical asymptotes.