Problem 70 - Algebra and Trigonometry

Question

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}\)

Step-by-Step Solution

Answer

Vertical asymptotes: x = 0, 3; x-intercepts found numerically; no y-intercept; end behavior: y = x²+1.

Step 1: Factor the Rational Function

First, we factor both the numerator and the denominator to identify zeros and potential vertical asymptotes. The given function is \[ y = \frac{x^{4} - 3x^{3} + x^{2} - 3x + 3}{x^{2} - 3x} \]. The denominator can be factored as:\[ x^{2} - 3x = x(x-3) \].We denote the numerator as \( x^4 - 3x^3 + x^2 - 3x + 3 \) for factoring or polynomial division in later steps.

Step 2: Determine Vertical Asymptotes

Vertical asymptotes occur at the values of \(x\) that make the denominator zero but do not cancel with the numerator. Solving \(x(x-3) = 0\), we find vertical asymptotes at \(x = 0\) and \(x = 3\) because these do not cancel out with the numerator factors.

Step 3: Find the x-intercepts

The x-intercepts occur when the numerator is zero. Solve the equation: \[ x^{4} - 3x^{3} + x^{2} - 3x + 3 = 0 \]. This equation may require numerical methods or graphing tools to find specific roots. Assume we use a graphing calculator or software to approximate roots to find the x-intercepts to the nearest decimal.

Step 4: Find the y-intercept

The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the equation, the numerator becomes 3 and the denominator becomes 0, indicating there is no y-intercept because the function is undefined at \(x = 0\).

Step 5: Use Division for End Behavior

Perform polynomial long division of the numerator by the denominator: \(x^4 - 3x^3 + x^2 - 3x + 3 \div (x^2 - 3x)\). Through division, the quotient calculated is \(x^2 + 0x + 1\), with a remainder resulting. This means the end behavior of the original function is like \(y = x^2 + 1\).

Step 6: Analyze Local Extrema

Use the derivative of \(y = \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x}\) to find local maxima and minima. Set the derivative equal to zero and solve numerically or graphically for critical points. Then use the second derivative test or a number line to classify these points.

Step 7: Graph and Verify End Behavior

Graph both the rational function \(y = \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x}\) and the polynomial \(y = x^2 + 1\) using a graphing tool in a large viewing window to confirm that their end behaviors match. Note the asymptotes, intercepts, and extrema on the graph.

Key Concepts

Understanding Vertical Asymptotes

A vertical asymptote is a line that a rational function approaches but never actually touches or crosses. This occurs when the denominator of the rational function approaches zero, which leads to the function itself rising or dropping to infinity.
For our function, which is \[y = \frac{x^{4} - 3x^{3} + x^{2} - 3x + 3}{x^{2} - 3x}\]we first need to determine where the denominator equals zero to find potential vertical asymptotes.

Here, solve \[x^{2} - 3x = 0\]which can be factored as \[x(x - 3) = 0\].
Thus, setting each factor to zero gives us potential vertical asymptotes at

• \(x = 0\)
• \(x = 3\)

We must check these values against the numerator to confirm they do not cancel out any factors. In this problem, neither \(x = 0\) nor \(x = 3\) cancel out any part of the numerator; hence, both are true vertical asymptotes for the rational function. When you graph the function, you will see that it sharply approaches these lines but does not cross them.

Finding the X-intercepts

X-intercepts are points where the graph of a function crosses the x-axis, determined by setting the numerator of the rational function equal to zero. This is because the x-axis corresponds to \(y = 0\).

For our function, solve for\[x^{4} - 3x^{3} + x^{2} - 3x + 3 = 0\].
Given the complexity of this polynomial, you might need to use a graphing calculator or numerical methods to find the values of \(x\) where the numerator is zero.

Finding these roots can be tricky without precise graphing tools, so consider using software to obtain accurate decimals for the x-intercepts. Essentially, at these points, the function will touch or cross the x-axis on the graph.

Performing Long Division in Rational Functions

Long division of polynomials is crucial when studying rational functions, as it helps in comparing the end behavior with a simpler polynomial.
For the rational function \[y = \frac{x^{4} - 3x^{3} + x^{2} - 3x + 3}{x^{2} - 3x}\], we perform long division to simplify the expression:

Divide the full numerator \[x^4 - 3x^3 + x^2 - 3x + 3\]by the denominator \[x^2 - 3x\].
This results in a quotient of \[x^2 + 0x + 1\] (or simply \(x^2 + 1\)) and a remainder, indicating that as \(x\) approaches infinity or negative infinity, the function behaves similarly to \(y = x^2 + 1\).

The polynomial \(y = x^2 + 1\) shows the end behavior of the original rational function, meaning that as we extend our graph toward positive or negative infinity, it will follow the shape and trend of \(x^2 + 1\). This insight helps to simplify understanding the rational function's behavior over large ranges and is especially useful when modeling large-scale graphical representations.