/*! 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 45 For the following exercises, gra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 45

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ k(x)=(x-3)^{3}(x-2)^{2} $$

Short Answer

Expert verified
The x-intercepts are at 3 and 2, with end behaviors (+鈭�, -鈭�) and y-intercept at -108.

Step by step solution

01

Identify the x-intercepts

The x-intercepts occur where the function equals zero. For \(k(x)=(x-3)^3(x-2)^2\), set each factor equal to zero: \(x-3=0\) and \(x-2=0\). Thus, the x-intercepts are \(x=3\) and \(x=2\).
02

Determine the multiplicity of each root

The multiplicity of a root is the exponent of the factor. For \((x-3)^3\), the root \(x=3\) has a multiplicity of 3. For \((x-2)^2\), the root \(x=2\) has a multiplicity of 2.
03

Determine the y-intercept

The y-intercept occurs where \(x=0\). Substitute \(x=0\) into the function: \(k(0)=(0-3)^3(0-2)^2=(-3)^3(-2)^2=-27 \times 4=-108\). Thus, the y-intercept is at \(y=-108\).
04

Analyze end behavior

The end behavior is determined by the leading term when the polynomial is expanded. The highest degree term is \(x^5\), which is positive, so as \(x\to\infty\), \(k(x)\to\infty\), and as \(x\to -\infty\), \(k(x)\to -\infty\).
05

Sketch the graph using key features

Plot the x-intercepts \(x=3\) and \(x=2\), noting that \(x=3\) has odd multiplicity (graph passes through) and \(x=2\) has even multiplicity (graph touches at the axis). Plot the y-intercept at \(y=-108\) and draw the end behavior as described in Step 4.

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.

Understanding x-intercepts
In polynomial functions, x-intercepts are the points where the graph crosses or touches the x-axis. These intercepts are found by setting the polynomial equation equal to zero and solving for the values of x. For the polynomial function \(k(x)=(x-3)^3(x-2)^2\), we find the x-intercepts by solving the equations \(x-3=0\) and \(x-2=0\). This gives us the x-intercepts at \(x=3\) and \(x=2\).
  • At \(x=3\): The graph will either cross or touch the x-axis.
  • At \(x=2\): Similarly, the graph will interact with the x-axis.
Finding x-intercepts is like identifying the points where the graph meets the horizontal ground. It's a key step in sketching the graph of a polynomial.
What is multiplicity?
Multiplicity refers to the number of times a particular x-intercept is repeated. It is indicated by the exponent of the factor in the polynomial. In \(k(x)=(x-3)^3(x-2)^2\), each factor's exponent tells us the multiplicity:
  • \(x=3\) has multiplicity 3: This means the graph will cross the x-axis at \(x=3\), creating a visible curve due to the odd multiplicity.
  • \(x=2\) has multiplicity 2: As this is an even multiplicity, the graph will "bounce" off the x-axis at \(x=2\), touching but not crossing.
The concept of multiplicity helps understand how the graph behaves at each x-intercept, providing insight into whether the graph will simply touch or actually cross the x-axis.
Finding the y-intercept
The y-intercept occurs where the graph crosses the y-axis, or when \(x=0\). To find the y-intercept of \(k(x)\), we substitute \(x=0\) in the function: \[(0-3)^3(0-2)^2 = (-3)^3(-2)^2 = -27 \times 4 = -108.\]
Thus, the y-intercept is at \(y=-108\).
  • The y-intercept is a useful point to plot, helping anchor the graph vertically.
  • It shows the function's value when all terms linked with x are at their simplest form.
The y-intercept provides an important fixed point about which the overall form of the polynomial graph is constructed.
Examining end behavior
The end behavior of a polynomial function describes how the graph behaves as \(x\) approaches positive or negative infinity. For \(k(x)=(x-3)^3(x-2)^2\), the leading term, which is derived from multiplying all the highest degree terms of each factor, is \(x^5\).
  • The degree (5) being odd and the leading coefficient being positive means the graph will rise to infinity as \(x\) approaches positive infinity.
  • Conversely, as \(x\) approaches negative infinity, the graph will fall to negative infinity.
Understanding end behavior gives a wide perspective on how the function behaves in larger scopes, ensuring the sketch reflects accurate trends at the edges of the graph.

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

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given. $$ f(x)=x^{3}+x-2, y=0,1,2 $$

For the following exercises, determine the function described and then use it to answer the question. The volume of a cylinder, \(V\), in terms of radius, \(r\), and height, \(h,\) is given by \(V=\pi r^{2} h .\) If a cylinder has a height of 6 meters, express the radius as a function of \(V\) and find the radius of a cylinder with volume of 300 cubic meters.

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=1-x^{3} $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{3}{x-4} $$

For the following exercises, use Kepler's Law, which states that the square of the time, \(T,\) required for a planet to orbit the Sun varies directly with the cube of the mean distance, \(a\), that the planet is from the Sun. Using Earth's distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.