/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Graph the rational function, and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 85

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest tenth. 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. $$r(x)=\frac{x^{4}-3 x^{3}+6}{x-3}$$

Short Answer

Expert verified
Vertical asymptote at \(x = 3\); x-intercepts around 0.6 and 2.4; y-intercept \(-2\). End behavior like \(x^3\).

Step by step solution

01

Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is equal to zero, as long as the numerator is not zero at the same point. The denominator of the function is \(x - 3\). Setting this equal to zero gives \(x = 3\). Thus, the vertical asymptote is at \(x = 3\).
02

Find X-Intercepts

X-intercepts occur where the numerator of the rational function is zero. Set \(x^4 - 3x^3 + 6 = 0\) and solve for \(x\). Finding exact roots algebraically might be complex, so numerical methods or graphing calculators can be used to approximate solutions. For this example, approximate solutions are around \(x \approx 0.6\) and \(x \approx 2.4\).
03

Find Y-Intercept

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function: \( r(0) = \frac{0^4 - 3(0)^3 + 6}{0 - 3} = -2 \). Thus, the y-intercept is at \( y = -2 \).
04

Perform Long Division for End Behavior

Divide the numerator \(x^4 - 3x^3 + 6\) by the denominator \(x - 3\) using polynomial long division. 1. Divide the leading term: \(x^4 / x = x^3\).2. Multiply \(x^3\) by \(x - 3\): \(x^4 - 3x^3\).3. Subtract \(x^4 - 3x^3\) from the original polynomial: remainder is \(6\).4. The result of the division is \(x^3\), and the remainder contributes \(\frac{6}{x-3}\).The resulting polynomial, which has the same end behavior as the rational function is \(x^3\).
05

Find Local Extrema

To find the local extrema, take the derivative of \(r(x)\), and find its critical points. However, as the division results suggest that the rational function approaches \(x^3\), we can analyze \(x^3\) for extrema.1. \(r'(x) = 3x^2\).2. Critical points are where \(r'(x) = 0\), i.e., \(x = 0\). At this point, evaluate the second derivative or test intervals around the point can show there are no local extrema between intervals influenced by division.
06

Graphing the Functions

Graph both the rational function \(r(x) = \frac{x^4 - 3x^3 + 6}{x - 3}\) and the polynomial \(x^3\) over a domain that includes values from -10 to 10. Observe that as \(x\) approaches positive or negative infinity, both functions approach a behavior similar to \(x^3\), confirming identical end behaviors. Additionally, include the vertical asymptote at \(x = 3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Vertical Asymptotes
Vertical asymptotes in a rational function indicate where the graph tends to infinity due to the denominator becoming zero. For the function \[ r(x) = \frac{x^4 - 3x^3 + 6}{x - 3} \] we identify vertical asymptotes by setting the denominator equal to zero.
  • The denominator is \( x - 3 \), set this equal to zero to find the vertical asymptote: \( x - 3 = 0 \).
  • Solving gives \( x = 3 \), indicating a vertical asymptote at \( x = 3 \).
Here, to avoid confusing a point of removable discontinuity, ensure the numerator does not also equal zero at \( x = 3 \). Since the numerator \( x^4 - 3x^3 + 6 \) is not zero at this point, \( x = 3 \) is a true vertical asymptote. Remind yourself to carefully factor the expression when necessary to detect any cancelable factors.
Polynomial Long Division
Polynomial long division is a method to simplify rational functions or find their end behavior. It鈥檚 similar to traditional long division but applied to polynomials. For the rational function \[ r(x) = \frac{x^4 - 3x^3 + 6}{x - 3} \], we want to divide the numerator by the denominator:
  • First, divide \( x^4 \) by \( x \), which gives us \( x^3 \).
  • Multiply \( x^3 \) by \( x - 3 \), resulting in \( x^4 - 3x^3 \).
  • Subtract this from the original numerator: \( x^4 - 3x^3 + 6 - (x^4 - 3x^3) \), leaving the remainder \( 6 \).
The polynomial division results give us \( x^3 \) plus the remainder \( \frac{6}{x-3} \). The polynomial \( x^3 \) provides a simplified view of the function's end behavior as it gives insight into what the function behaves like as \( x \to \infty \). Remember, it鈥檚 essential to correctly compute the remainder when dividing to maintain accuracy.
Local Extrema
Local extrema are points where a function reaches a local maximum or minimum. To find these for a rational function, examine the derivative to detect critical points.
  • For \( r(x) \), derive the function to get \( r'(x) \).
  • The critical points of the function occur where \( r'(x) = 0 \).
Using the simplified polynomial \( x^3 \) reaches its derivative as \( 3x^2 \). Solving \( 3x^2 = 0 \) gives \( x = 0 \). Evaluating the second derivative \( r''(x) = 6x \) shows that at \( x = 0 \), the second derivative is zero indicating neither a local maximum nor minimum is present. It鈥檚 important while using derivatives that the manipulations and simplifications correspond with real behavior in intervals to not miss any additional extrema the rational function has.
X-Intercepts
X-intercepts of a function occur where the output, or the y-value, is zero. For rational functions, this corresponds to where the numerator is zero, but not the denominator.
  • For \( r(x) = \frac{x^4 - 3x^3 + 6}{x - 3} \), find \( x \) such that \( x^4 - 3x^3 + 6 = 0 \).
  • Solving polynomial equations can be cumbersome, so tools like graphing calculators might be necessary to approximate solutions.
  • Approximated solutions provide x-intercepts at around \( x \approx 0.6 \) and \( x \approx 2.4 \).
Approximations should be verified if possible by substituting back into the equation or cross-referencing graphically. A thorough understanding of polynomial roots and stepwise checking will ensure accuracy.
Y-Intercepts
The y-intercept is the point where the graph of the function intersects the y-axis, which happens when \( x = 0 \). For rational functions:
  • Substitute \( x = 0 \) into the function \( r(x) \).
  • For \[ r(x) = \frac{x^4 - 3x^3 + 6}{x - 3} \], substituting gives \[ r(0) = \frac{0^4 - 3(0)^3 + 6}{0 - 3} = -2 \].
Thus, the y-intercept for this function is \( y = -2 \). Understanding this concept helps you pinpoint the start of graphing the function on a Cartesian plane, giving you a reference point to align other graphical features.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A highway engineer develops a formula to estimate the number of cars that can safely travel a particular highway at a given speed. She finds that the number \(N\) of cars that can pass a given point per minute is modeled by the function $$N(x)=\frac{88 x}{17+17\left(\frac{x}{20}\right)^{2}}$$ Graph the function in the viewing rectangle \([0,100]\) by \([0,60] .\) If the number of cars that pass by the given point is greater than \(40,\) at what range of speeds can the cars travel?

Find the factors that are common in the numerator and the denominator. Then find the intercepts and asymptotes, and sketch a graph of the rational function. State the domain and range of the function. $$r(x)=\frac{x^{2}-2 x-3}{x+1}$$

Graphing Quadratic Functions A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Find the vertex and \(x\) and \(y\) -intercepts of \(f .\) (c) Sketch a graph of \(f .\) (d) Find the domain and range of \(f\). $$f(x)=2 x^{2}+12 x+10$$

Find the maximum or minimum value of the function. $$h(x)=\frac{1}{2} x^{2}+2 x-6$$

Find the factors that are common in the numerator and the denominator. Then find the intercepts and asymptotes, and sketch a graph of the rational function. State the domain and range of the function. $$r(x)=\frac{x^{3}-2 x^{2}-3 x}{x-3}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.