Problem 74 - Algebra and Trigonometry

Question

Graph the rational function \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, g(x)=-x+4 $$

Step-by-Step Solution

Answer

Vertical asymptotes are at \(x=0\) and \(x=2\); both functions have similar end behavior.

Step 1: Identify Asymptotes

To find vertical asymptotes, we need to determine where the denominator of \( f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x} \) equals zero. Set \(x^2 - 2x = 0\) and solve for \(x\). This factors to \( x(x - 2) = 0 \), giving vertical asymptotes at \( x = 0 \) and \( x = 2 \).

Step 2: Simplify the Function

To simplify \( f(x) \), perform polynomial division or factor both the numerator and denominator. The denominator \( x^2 - 2x \) can be factored as \( x(x - 2) \), and the numerator \( -x^3 + 6x^2 - 5\) does not have a common factor that simplifies \( f(x) \) further in a meaningful way for finding end behavior.

Step 3: Determine End Behavior

Check the leading term of each function to determine their end behavior. For \( f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x} \), as \( x \to \pm \infty \), the dominant terms are \( -x^3 \) in the numerator and \( x^2 \) in the denominator, so \( f(x) \sim -x \). The function \( g(x) = -x + 4 \) also has a leading term of \( -x \), indicating \( f(x) \) and \( g(x) \) share the same asymptotic behavior at both ends.

Step 4: Graph Functions

Use graphing technology or sketch by plotting points. Graph \( f(x) \) and \( g(x) \) in a viewing window large enough to include \( x \) values like \( -10 \) to \( 10 \) to observe the end behavior. Note the positions where \( f(x) \) has vertical asymptotes at \( x = 0 \) and \( x = 2 \), and observe that both functions approach lines with similar slopes (asymptotic lines) proving similar behavior.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes occur where the function is undefined, often when the denominator equals zero. For a function like \( f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\), finding where the denominator is zero helps locate these asymptotes.
Start by setting the denominator equal to zero: \( x^2 - 2x = 0 \). Factoring gives us \( x(x - 2) = 0 \,\) which provides the roots \( x = 0 \) and \( x = 2 \,\). These are the points where vertical asymptotes exist.

• Vertical asymptotes indicate values that the function will approach but never reach.
• They might split the graph into sections where the function heads toward positive or negative infinity as it nears the asymptote from either side.

When graphing, remember that the presence of a vertical asymptote suggests the graph will not cross this line. It's essential for analyzing rational functions and understanding their behavior in graphical form.

Polynomial Division

Polynomial division is a method used to simplify complex rational functions, similar to long division with numbers. In the case of \( f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\), using polynomial division might seem tempting, but here, it isn't necessary for simplification.
Often, you can try to factor both the numerator and the denominator to see if common factors exist. For our function, the denominator factors easily into \( x(x - 2) \), but the numerator \( -x^3 + 6x^2 - 5 \) doesn't simplify significantly in terms of removable factors.

• Polynomial division is still useful when dealing with more complicated polynomials where direct simplification isn't apparent.
• This technique can also assist in reducing the function to a more easily analyzed form, especially when finding the end behavior or sketching the graph.

In this particular exercise, despite our attempt, polynomial division doesn’t lead to further simplification for the end behavior analysis.

End Behavior

The end behavior of a rational function examines what happens to the function's value as \( x \) approaches positive or negative infinity. For both \( f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\) and \( g(x) = -x + 4\), it's useful to look at the leading terms.
In \( f(x),\) the dominant terms are \(-x^3\) in the numerator and \( x^2\) in the denominator. As you let \( x \) approach infinity, the function \( f(x) \) behaves similar to \( -x \,\) just like the function \( g(x) = -x + 4.\)

• For both functions, as \( x \) becomes very positive or very negative, their values mimic those of \( -x \).
• This indicates that the functions have the same slope and general direction at extreme values, even if their behavior in between differs due to other terms.

Understanding end behavior is crucial because it shows how the function acts beyond the immediate neighborhood of special points like intercepts and asymptotes, giving a full picture of the function's long-term trends.