/*! 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 84 For each polynomial, a. find the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 84

For each polynomial, a. find the degree; b. find the zeros, if any; c. find the \(y\) -intercept(s), if any; d. use the leading coefficient to determine the graph's end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. \(f(x)=-3 x^{2}+6 x\)

Short Answer

Expert verified
The degree is 2, zeros are 0 and 2, the y-intercept is 0, both ends go downward, and it is neither even nor odd.

Step by step solution

01

Find the Degree

To find the degree of the polynomial, look for the highest power of the variable. Here, the polynomial is \( f(x) = -3x^2 + 6x \). The highest power of \( x \) is 2, so the degree of the polynomial is 2.
02

Find the Zeros

To find the zeros of the polynomial, we solve the equation \( f(x) = 0 \). Set \( -3x^2 + 6x = 0 \). Factor out the common term \( -3x \), giving \( -3x(x - 2) = 0 \). This gives the zeros \( x = 0 \) and \( x = 2 \).
03

Find the Y-intercept

The \( y \)-intercept is the value of the function when \( x = 0 \). Substitute \( x = 0 \) into the polynomial: \( f(0) = -3(0)^2 + 6(0) = 0 \). Therefore, the \( y \)-intercept is 0.
04

Determine End Behavior

The leading coefficient of the polynomial is the coefficient of the term with the highest power of \( x \), in this case, \(-3\). Since the degree is 2 (an even number) and the leading coefficient is negative, both ends of the graph will point downwards as \( x \to -\infty \) and \( x \to \infty \).
05

Determine If Polynomial is Even, Odd, or Neither

A polynomial is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \). Evaluate \( f(-x) = -3(-x)^2 + 6(-x) = -3x^2 - 6x \). Since \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), the polynomial is neither even nor odd.

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.

Degree of a Polynomial
The degree of a polynomial is determined by the highest power of the variable present in the expression. In this instance, the polynomial given is \( f(x) = -3x^2 + 6x \).
When examining this polynomial, you will notice that the term with the highest power of \( x \) is \( x^2 \). Therefore, the degree of this polynomial is 2. The degree plays a significant role in defining the properties and behavior, such as the number of zeros it can have and the shape of its graph.
Zeros of a Function
Zeros, also known as roots, of a polynomial, are the values of \( x \) for which the function equals zero. Finding zeros is essential for understanding where the graph of the polynomial will intersect the \( x \)-axis. To find the zeros of our polynomial \( f(x) = -3x^2 + 6x \), set the equation to zero: \(-3x^2 + 6x = 0\).
Factor out the common term \( -3x \) to simplify the expression: \(-3x(x - 2) = 0\).
This gives us two zeros: \( x = 0 \) and \( x = 2 \).
These are the points where the polynomial crosses the \( x \)-axis.
Y-intercept
The \( y \)-intercept of a function is the point at which the graph intersects the \( y \)-axis. It occurs when the value of \( x \) is zero. To find the \( y \)-intercept of the polynomial \( f(x) = -3x^2 + 6x \), substitute \( x = 0 \) into the function: \( f(0) = -3(0)^2 + 6(0) = 0 \).
This means the \( y \)-intercept is at the origin, \( (0, 0) \), which is a key point in graphing and understanding the behavior of the polynomial.
End Behavior
End behavior describes how the graph of a polynomial behaves as the values of \( x \) approach positive or negative infinity. It is influenced by the leading term of the polynomial, which is the term with the highest power of \( x \) and its coefficient. In the polynomial \( f(x) = -3x^2 + 6x \), the leading term is \(-3x^2\).
This polynomial has an even degree (2) and a negative leading coefficient. Because of these properties, both ends of the graph will "point" downwards:
  • As \( x \to -\infty \), \( f(x) \to -\infty \)
  • As \( x \to \infty \), \( f(x) \to -\infty \)
Understanding end behavior can help predict how the graph will look, especially outside the visible range.
Even and Odd Functions
Determining whether a polynomial is even, odd, or neither helps in predicting symmetries in its graph. For a function \( f(x) \):
  • It is even if \( f(-x) = f(x) \). This indicates symmetry about the \( y \)-axis.
  • It is odd if \( f(-x) = -f(x) \), indicating rotational symmetry about the origin.
To determine the nature of the given polynomial \( f(x) = -3x^2 + 6x \), calculate \( f(-x) \): \( f(-x) = -3(-x)^2 + 6(-x) = -3x^2 - 6x \).
Notice that \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \).
This means the polynomial is neither even nor odd.

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 problems, consider the population of Ocean City, New Jersey, which is cyclical by season. The population can be modeled by \(P(t)=82.5-67.5 \cos ((\pi / 6) t],\) where \(t\) is time in months \((t=0 \text { represents January } 1)\) and \(P\) is population (in thousands). During a year, in what intervals is the population less than \(20,000 ?\) During what intervals is the population more than \(140,000 ?\)

Solve the following trigonometric equations on the interval \(\theta=[-2 \pi, 2 \pi]\) exactly. $$ \sec ^{2} x-2 \sec x+1=0 $$

For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. $$ g(x)=\left\\{\begin{array}{l}{\frac{3}{x-2}, x \neq 2} \\ {4, x=2}\end{array} ; g(0) ; g(-4) ; g(2)\right. $$

For the following exercises, use the change-of-base formula and either base 10 or base \(e\) to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. $$ \log _{6} 103 $$

ITI The admissions office at a public university estimates that 65\(\%\) of the students offered admission to the class of 2019 will actually enroll. a. Find the linear function \(y=N(x),\) where \(N\) is the number of students that actually enroll and \(x\) is the number of all students offered admission to the class of 2019 . b. If the university wants the 2019 freshman class size to be 1350 , determine how many students should be admitted.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.