Problem 78 - 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 at \(x=0\) and \(x=3\); polynomial \(x^2 - 3\) matches end behavior.

Step 1: Simplify the Rational Function

The expression is given as \( y=\frac{x^{4}-3x^{3}+x^{2}-3x+3}{x^{2}-3x} \). Start by factoring the denominator: \(x^2 - 3x = x(x-3)\). Next, consider factoring the numerator or using polynomial division. For simplification here, since no factoring simplifies well, we directly move to polynomial division to find end behavior.

Step 2: Perform Polynomial Long Division

Divide \(x^4 - 3x^3 + x^2 - 3x + 3\) by \(x^2 - 3x\). First term: \(x^2\), then subtract to get \(x^2 - 3x - 3\). Next term: this also gives \(1\), final remainder \(-3\). Thus, the division gives: \[x^2 - 3 + \frac{-3}{x(x-3)}\]. The polynomial \(x^2 - 3\) reflects the end behavior.

Step 3: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at the same point. Set \(x(x-3) = 0\), giving \(x = 0\) and \(x = 3\) as vertical asymptotes.

Step 4: Find the x-intercepts

The x-intercepts occur where the function is zero, i.e., the numerator equals zero. Since solving \(x^4 - 3x^3 + x^2 - 3x + 3 = 0\) is complex remaining then without obvious factors, only numerical approximation or graphing would show specific x-intercepts.

Step 5: Find the y-intercept

The y-intercept is found by evaluating the function at \(x=0\). Substitute into the original equation: \(y = \frac{0^4-3*0^3+0^2-3*0+3}{0^2-3*0} = \frac{3}{0}\), undefined at this point, indicating x-intercept alteration due to asymptote.

Step 6: Evaluate Local Extrema

Compute the derivative of the function to find critical points, solving through modification given the complexity here or using tools such as graphing calculators to determine local extrema or turning points on the graph directly.

Step 7: Verify End Behaviors via Graphs

Graph \(y = \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x}\) and \(y = x^2 - 3 \) in large windows to observe end behavior matching. The polynomial accounts for how the original function diverges as \(x\) approaches infinity.

Key Concepts

Graphing Rational Functions

A rational function is essentially a fraction where the numerator and the denominator are both polynomials. Graphing rational functions involves identifying key features such as intercepts, asymptotes, and end behavior. When graphing, start by:

• Identifying the x- and y-intercepts
• Determining any vertical, horizontal, or slant asymptotes
• Checking the overall end behavior

Graphing tools, such as graphing calculators or software, can provide visual understanding. When you represent a rational function graphically, it often has sections that approach vertical asymptotes and curve towards end behavior directions.

Vertical Asymptotes

Vertical asymptotes occur at the points where the denominator of a rational function equals zero, unless those points are canceled out by similar factors in the numerator. These are essentially boundaries that the graph cannot cross; the function approaches infinity.In our function, the denominator is \(x^2 - 3x = x(x-3)\). Setting this equal to zero, we solve for \(x = 0\) and \(x = 3\), giving the vertical asymptotes. The graph will tend to go upwards or downwards sharply at these values of \(x\).

End Behavior

End behavior describes how the function behaves as \(x\) approaches positive or negative infinity. For rational functions, this often involves examining the degrees of the polynomials in the numerator and denominator.We use long division to simplify the function and match it to a polynomial. In our case, dividing gives \(x^2 - 3\). This polynomials' terms dictate that as \(x\) heads towards infinity in both directions, the rational function will behave similarly to \(x^2 - 3\). This represents a parabolic curve opening upward, reflecting the main component of the original function's shape at extremes.

Long Division of Polynomials

Long division helps simplify complex rational functions. It helps determine the end behavior by breaking down the fraction into a simpler polynomial and a remainder. For the function \(y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}\), division reveals \(x^2 - 3\) with a remainder. The simplified form matches how the graph reflects behavior at large values of \(x\). This process is crucial as it offers insight into how complicated functions mimic simpler polynomial ones at their extremes.

Intercepts of Rational Functions

Intercepts are the points where the graph crosses the axes. The **x-intercepts** occur when the numerator is zero, leading the entire function to zero. Solutions to \(x^4 - 3x^3 + x^2 - 3x + 3 = 0\) will yield these intercepts. Without clear factors, numerical methods or graph analysis is necessary for finding specific values.The **y-intercept** occurs when \(x = 0\). Substituting, however, reveals it is undefined due to a division by zero, related to the vertical asymptote at \(x = 0\). Thus, the function does not cross the y-axis but approaches near it.