Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{x^{2}-4 x+3}{x^{2}-4 x-5} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Local Behavior

To analyze the local behavior of a rational function, we look for critical points by considering the derivative or solving for values that make the function or its denominator equal to zero. - For the function \( f(x)= \frac{x^{2}-4x+3}{x^{2}-4x-5} \), factor both the numerator and the denominator.- The numerator \( x^2 - 4x + 3 = (x-1)(x-3) \).- The denominator \( x^2 - 4x - 5 = (x-5)(x+1) \).- Check for shared factors: There are none, so no removable discontinuities.- The function has vertical asymptotes at \( x = 5 \) and \( x = -1 \) and zeros at \( x = 1 \) and \( x = 3 \).

Step 2: Determine End Behavior

The end behavior of rational functions is determined by the leading terms when the degree of the numerator and denominator are compared.- Both the numerator and the denominator have the degree 2, indicating the function behaves like \( \frac{1}{1} \cdot x^2 \), as \( x \to \infty \) or \( -\infty \).- Thus, as \( x \to \infty \) or \( x \to -\infty \), the function approaches \( f(x) = 1 \). This means there is a horizontal asymptote at \( y = 1 \).

Key Concepts

Local Behavior

The local behavior of a function gives us insights into how the function behaves at specific points or near certain inputs. For a rational function like \( f(x) = \frac{x^{2} - 4x + 3}{x^{2} - 4x - 5} \), identifying the local behavior involves factoring both the numerator and the denominator. By doing so, we locate the zeros of the function and the values that make the denominator zero, helping us understand where the function has critical points such as zeros or poles.

• First, factor the numerator: \( x^2 - 4x + 3 = (x-1)(x-3) \).
• Next, factor the denominator: \( x^2 - 4x - 5 = (x-5)(x+1) \).

The zeros of the function occur where the numerator equals zero, excluding where the denominator is zero, giving us zeros at \( x = 1 \) and \( x = 3 \). There are no shared factors between the numerator and the denominator, indicating no removable discontinuities or holes. However, where the denominator is zero (at \( x = 5 \) and \( x = -1 \)), the function is undefined, leading to vertical asymptotes.

End Behavior

End behavior describes how a function behaves as the input values approach infinity or negative infinity. For rational functions, the end behavior is mainly influenced by the degrees of their polynomials in both the numerator and the denominator. In our function \( f(x) = \frac{x^{2} - 4x + 3}{x^{2} - 4x - 5} \), both the numerator and the denominator are of degree 2.
When the degrees of the numerator and the denominator are equal, the function approaches a horizontal asymptote determined by the ratio of the leading coefficients. Here, this ratio is \( \frac{1}{1} \), hence as \( x \to \infty \) or \( x \to -\infty \), the function approaches \( f(x) = 1 \). This informs us that there is a horizontal asymptote at \( y = 1 \). Knowing the end behavior helps us sketch the graph or predict how outputs behave for very large or very small inputs.

Vertical Asymptotes

Vertical asymptotes in rational functions occur where the function's denominator equals zero, as long as these points do not coincide with zeros in the numerator. In \( f(x) = \frac{x^{2} - 4x + 3}{x^{2} - 4x - 5} \), we've found the denominator factors to \( (x-5)(x+1) \).

• Setting these factors to zero gives \( x = 5 \) and \( x = -1 \), which are the locations of vertical asymptotes.
• At a vertical asymptote, the function will increase or decrease without bound, hence the graph will make a sharp turn upwards or downwards at these points.

These asymptotes divide the graph into sections where the function will behave differently on either side of the asymptotes. It's important to note that as \( x \) approaches these values, the function \( f(x) \) becomes undefined and the outputs move towards positive or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes help describe the behavior of functions at the extremities of the graph, meaning as \( x \to \infty \) or \( x \to -\infty \). They are important for understanding how a function settles into a predictable value if it does so. For our function \( f(x) = \frac{x^{2} - 4x + 3}{x^{2} - 4x - 5} \), the degrees of the polynomials in the numerator and the denominator are the same: both are degree 2.
For functions where the degrees are equal, a horizontal asymptote can be found by dividing the leading coefficients. Here, since both leading terms are \( x^2 \), the coefficients are both 1, yielding a horizontal asymptote at \( y = 1 \).
Having a horizontal asymptote at \( y = 1 \) means that as the x-values increase or decrease significantly, the value of the function approaches 1. The presence of a horizontal asymptote doesn’t necessarily mean the function will never cross this line, but generally provides a value the function will not deviate from significantly as input becomes large.