Problem 74 - Precalculus Mathematics for Calculus

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. $$f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, \quad g(x)=-x+4$$

Step-by-Step Solution

Answer

Vertical asymptotes of \(f(x)\) are \(x = 0\) and \(x = 2\). Both \(f(x)\) and \(g(x)\) share the same end behavior, following the line \(-x\).

Step 1: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is zero and the numerator is not zero at the same point. Set the denominator of \(f(x)\), \(x^2 - 2x = 0\), equal to zero and solve for \(x\). Factoring gives \(x(x - 2) = 0\). Thus, the solutions are \(x = 0\) and \(x = 2\). These are the vertical asymptotes of \(f(x)\).

Step 2: Determine End Behavior of f(x)

The end behavior of a rational function is determined by the leading terms of its numerator and denominator. In \(f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\), as \(x\) approaches infinity or negative infinity, the term \(-x^3\) in the numerator will dominate, and the denominator behaves as \(x^2\). Thus, the end behavior can be described by the quotient of the leading terms, \(-x^3/x^2 = -x\). As \(x\to\infty\), \(f(x) o -x\).

Step 3: Compare End Behavior with g(x)

The end behavior of \(g(x) = -x + 4\) as \(x\to\infty\) is also dominated by the term \(-x\). Therefore, \(f(x)\) and \(g(x)\) have the same end behavior, both approaching \(-x\) as \(x\to\infty\) or \(x\to -fty\).

Step 4: Graphing f(x) and g(x)

In a large enough viewing rectangle, graph \(f(x)\) showing the vertical asymptotes at \(x = 0\) and \(x = 2\). Graph \(g(x) = -x + 4\) on the same plot. Notice that both functions exhibit similar end behavior by appearing to approach the same general trend line as \(x\to\infty\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines on a graph that correspond to the values of \(x\) where a rational function becomes undefined. In simpler terms, they are where the graph shoots up to infinity or dips down to negative infinity. Finding these helps in understanding the behavior of the function at certain points.

For the function \(f(x) \), vertical asymptotes occur where the denominator is zero but the numerator is not zero simultaneously. This happens because division by zero is undefined in mathematics.

Let's look at the function \(f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\). You set the denominator, \(x^2 - 2x\), to zero to find these points.

• Solve \(x(x - 2) = 0\) to get \(x = 0\) and \(x = 2\).
• These values indicate the vertical asymptotes of \(f(x)\).

On the graph, you'll see the curves reaching towards infinity or negative infinity at these \(x\) values.

End Behavior

End behavior describes what happens to the values of a function \(f(x)\) as \(x\) becomes very large (positively or negatively). Understanding end behavior is key to anticipating how functions act at extreme values, giving you a clearer picture of the function's trend.

For rational functions, this involves the leading terms of the numerator and denominator. In \(f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\), as \(x\) heads towards infinity, the dominant term is \(-x^3\) in the numerator and \(x^2\) in the denominator.

Simplifying the behavior to the quotient\(-x^3/x^2 = -x\), you find that as \(x\) approaches infinity or negative infinity, \(f(x)\) will resemble the line \(y = -x\). Thus, \(f(x)\) tends toward \(-x\) on both ends.

Similarly, for \(g(x) = -x + 4\), the term \(-x\) prominently influences its end behavior.

Graphing Functions

Graphing is a powerful way to visualize and analyze functions. It transforms abstract equations into visible and understandable shapes. For \(f(x)\) and \(g(x)\), plotting becomes crucial in observing both vertical asymptotes and end behaviors efficiently.

When graphing \(f(x)\) \(= \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\):

• Mark the vertical asymptotes \(x = 0\) and \(x = 2\). These are essential features, showing where the function is not defined and behaves unstably.
• Notice the curves of the graph approaching these vertical lines but never crossing over or touching them.

For \(g(x) = -x + 4\), plot the straight line which acts as a baseline against which \(f(x)\)'s end behavior can be compared.

In a large enough viewing area, both functions will appear to converge towards similar directions in terms of their end behavior as \(x\) grows larger in either direction.

Polynomials

Polynomials are expressions that use addition, subtraction, and multiplication involving variables. They are foundational in understanding more complex functions like rational functions.

Here, the polynomial components of \(f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}\) guide its overall shape and behavior.

• The numerator \(-x^3 + 6x^2 - 5\) and the denominator \(x^2 - 2x\) are polynomials that help determine key characteristics of \(f(x)\) such as its degree and leading terms.
• The degree of a polynomial is the highest power of the variable \(x\). For \(f(x)\), the numerator has a degree of 3, and the denominator has a degree of 2.

Understanding polynomials helps in predicting behaviors like growth rate and the general shape of the graph. Recognizing these can simplify the interpretation of the larger, more complex rational functions, making math less intimidating and more approachable.