Question

In Exercises \(39-44,\) (a) find a power function end behavior model for \(f .\) (b) Identify any horizontal asymptotes. $$f(x)=\frac{x-2}{2 x^{2}+3 x-5}$$

Short answer

The power function end behavior model for the function \(f(x)=\frac{x-2}{2 x^{2}+3 x-5}\) is \(y = 0\). The horizontal asymptote of this function is also \(y = 0\).

Step-by-step solution

Identify the Degree of Numerator and Denominator

For the function \(f(x)=\frac{x-2}{2 x^{2}+3 x-5}\), the degree of the polynomial in the numerator (which is \(x\)) is 1. The degree of the denominator (which is (\(2x^2+3x-5\)) is 2.

Find the Power Function End Behavior Model

Since the degree of the denominator is greater than the degree of the numerator, the end behavior model is \(y = 0\). This is because as x becomes very large (approaches infinity), the \(2x^{2}\) in the denominator dominates the denominator, and similarly, the \(x\) term in the numerator dominates the numerator. Therefore, the overall fraction approaches zero, thus making \(y = 0\) the end behavior model.

Identify Horizontal Asymptotes

A horizontal asymptote is formed due to the end behavior of a function. For this function, as concluded from Step 2, the horizontal asymptote is \(y = 0\). That is, as \(x\) approaches positive or negative infinity, the function \(f(x)\) tends towards 0. This shows that the line \(y = 0\) is a horizontal asymptote.

Key concepts

Power Function

A power function is one of the simplest and most important types of functions in algebra. It is generally represented in the form of \( f(x) = ax^n \), where \( a \) is a constant, \( n \) is a non-negative integer, and \( x \) is a variable. The constant \( a \) is called the coefficient, and \( n \) is referred to as the exponent or the degree of the function. The behavior of power functions varies significantly with the value of the exponent. When analyzing the end behavior of power functions, that is, how the function behaves as \( x \) approaches infinity or negative infinity, the degree of the function plays a crucial role. If the degree is even, the end behavior will see the function’s value rise or fall towards infinity, depending on the sign of the coefficient. If the degree is odd, one tail of the function will head towards infinity while the other falls towards negative infinity, again depending on the sign of the coefficient.

Power functions with higher degrees grow faster than those with lower degrees. For example, a function with a degree of 3 (cubic) will grow faster than a function with a degree of 1 (linear) as \( x \) becomes very large. This important characteristic of power functions is essential in determining the end behavior of more complex functions, such as rational functions, where the degree of the numerator and denominator must be considered to predict the behavior at the extremities of the graph.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches, but never actually reaches, as the independent variable (usually \( x \)) heads towards positive or negative infinity. To find a horizontal asymptote, you often compare the degrees of the numerator and denominator in a rational function— which is a ratio of two polynomials. If the degree of the numerator is less than the degree of the denominator, like we see in the exercise with \( f(x) = \frac{x-2}{2x^{2}+3x-5} \), the horizontal asymptote will be the \( x \)-axis, or \( y=0 \).

If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote will be the ratio of the leading coefficients of both polynomials. Conversely, if the degree of the numerator is greater than the degree of the denominator, the function has no horizontal asymptote as it will grow without bound. Understanding horizontal asymptotes is crucial for sketching the long-term behavior of functions and is particularly useful for calculus, where functions can be analyzed for their limits at infinity.

Polynomial Degree

The degree of a polynomial is the highest power of the variable within its terms, assuming the polynomial is written in standard form. This means that in a polynomial like \( 4x^5 - 2x^3 + x - 7 \), the degree would be 5, which is the highest exponent of \( x \). The degree of a polynomial is significant because it provides information about the possible number of roots, the shape of the graph, and the behavior of the polynomial at extremes. It also helps in determining end behavior, as seen in our exercise.

The degree of the polynomial in the numerator of the function \( f(x) \) is 1, while that of the denominator is 2. Since the degree of the denominator is higher, as per the aforementioned relationship, the function has the horizontal asymptote of \( y = 0 \) and an end behavior model where the function approaches zero as \( x \) becomes larger. The degree of a polynomial is a vital concept to master because it leads to an understanding of polynomial functions broadly and can be applied to analyzing and graphing these functions accurately.