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Chapter 3: Problem 70

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. $$y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}$$

Short Answer

Expert verified
Vertical asymptotes: \(x = 0\) and \(x = 3\); Polynomial end behavior: \(x^2 - 3x + 1\).

Step by step solution

01

Identify Vertical Asymptotes

To find the vertical asymptotes, set the denominator of the function equal to zero: \[ x^2 - 3x = 0 \]. Factor the equation: \[ x(x - 3) = 0 \]. This gives us the vertical asymptotes at \( x = 0 \) and \( x = 3 \), assuming these values do not lead to factors in the numerator that can cancel them.
02

Find the x-intercepts

To find the x-intercepts, set the numerator equal to zero: \[ x^4 - 3x^3 + x^2 - 3x + 3 = 0 \]. Use any method such as synthetic division, factoring, or numerical methods to approximate the roots of the numerator. Solving numerically or graphically may be necessary to find the x-intercepts, as the equation does not factor cleanly.The x-intercepts could be estimated based on a graph.
03

Find the y-intercept

To find the y-intercept, substitute \( x = 0 \) into the function: \[ y = \frac{0^4 - 3 \cdot 0^3 + 0^2 - 3 \cdot 0 + 3}{0^2 - 3 \cdot 0} \]. Simplifying gives us a division by zero, indicating a vertical asymptote rather than a y-intercept at \( x = 0 \). Therefore, there is no y-intercept since the function is not defined at \( x = 0 \).
04

Determine Local Extrema

Local extrema occur at critical points where the first derivative is zero or undefined. First, find the derivative \( y' \) and solve \( y' = 0 \). Given the complexity, use numerical methods to approximate values where \( y' = 0 \) and the function changes from increasing to decreasing, or vice versa. For simplicity, extremum points could also be estimated graphically.
05

Use Long Division for End Behavior

Perform polynomial long division on \( \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x} \) to find a polynomial function with equivalent end behavior. Dividing gives a quotient of \( x^2 - 3x + 1 \). The remainder becomes negligible as \( x \) approaches infinity, so the polynomial \( x^2 - 3x + 1 \) reflects the rational function's end behavior.
06

Graph the Functions

Graph the original rational function and the polynomial \( x^2 - 3x + 1 \) over a large domain such as \([-10, 10]\) in both the x and y directions. Observe how both graphs approach the same end behavior as \( x \rightarrow \pm \infty \). Use software or a graphing calculator for accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
Vertical asymptotes occur in the graph of a rational function where the denominator is equal to zero, but the numerator does not cancel this factor. To find the vertical asymptotes of our function:
  • Set the denominator equal to zero: \( x^2 - 3x = 0 \).
  • Factor this equation: \( x(x - 3) = 0 \).
  • Find the solutions for \( x \): these are \( x = 0 \) and \( x = 3 \).
These values make the denominator of the function zero, leading to a potential vertical asymptote unless the numerator is also zero at these points (which would lead to a hole instead). Here, neither \( x = 0 \) nor \( x = 3 \) cancel with the numerator, confirming that both are vertical asymptotes. As you approach these x-values from either direction on the graph, the y-values will approach positive or negative infinity.
x-intercepts
The x-intercepts of a function represent the points where the graph crosses the x-axis, meaning y equals zero. To find these:
  • Set the numerator of the rational function equal to zero: \( x^4 - 3x^3 + x^2 - 3x + 3 = 0 \).
  • Solve for \( x \), which can be complex since the equation does not factor easily.
Often, you will need to rely on numerical methods such as graphing calculators or software to approximate solutions. For practical purposes, you may graph the function to visually estimate where it crosses the x-axis. These are your x-intercepts and they can help guide you in sketching the basic shape of the graph.
y-intercepts
Finding the y-intercept of a function usually involves setting \( x=0 \). However, in this rational function, substituting \( x=0 \) results in division by zero.
  • The function is expressed as: \( y = \frac{0^4 - 3 \cdot 0^3 + 0^2 - 3 \cdot 0 + 3}{0^2 - 3 \cdot 0}\).
  • Because the denominator becomes zero, there is no y-intercept at \( x = 0 \).
This happens due to a vertical asymptote, meaning the graph approaches infinity rather than crossing the y-axis. Therefore, for this function, a traditional y-intercept does not exist, highlighting the importance of considering asymptotic behavior in graphing rational functions.
Local Extrema
Local extrema (i.e., maximum and minimum points of a graph) occur where the derivative is zero or undefined, indicating a change in direction of the slope of the function. To find these:
  • Calculate the first derivative of the function. Use calculus to do this accurately: derive \( y' \).
  • Set \( y' = 0 \) and solve for \( x \). These x-values are potential extrema points.
  • Use numerical methods or graphs to estimate these points, as the derivative may not allow easy algebraic solving.
Inspecting these points' neighborhoods will reveal whether they are maximums, minimums, or neither, based on whether the function moves from increasing to decreasing or vice versa. These help to better understand the local behavior of the graph.
Polynomial Long Division
Polynomial long division is a method used to divide polynomials, providing insight into the end behavior of rational functions. Here, it helps to find a polynomial with equivalent end behavior:
  • Divide the numerator \( x^4 - 3x^3 + x^2 - 3x + 3 \) by the denominator \( x^2 - 3x \).
  • The result is a quotient of \( x^2 - 3x + 1 \).
  • This quotient forms a polynomial representing the end behavior of the rational function.
As \( x \) approaches positive or negative infinity, the influence of the remainder diminishes, and the behavior of the rational function mirrors that of the polynomial \( x^2 - 3x + 1 \). Graphing both functions shows that they converge in behavior towards the ends of the graph, reflecting the same asymptotic trends.

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Most popular questions from this chapter

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. $$f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x+6}{-6 x+3}$$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=3 x^{4}-17 x^{3}+24 x^{2}-9 x+1 ; \quad a=0, b=6$$

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\). (b) Sketch the graph of \(P\). $$P(x)=-x^{3}-2 x^{2}+5 x+6$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{3}+4}{2 x^{2}+x-1}$$
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