Leading Coefficient Test - Varsity Tutors

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Leading Coefficient Test

Study Guide

Key Definition

The Leading Coefficient Test determines the end behavior of the graph of a polynomial function $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ based on the leading coefficient $a_n$ and the degree $n$.

Important Notes

  • If $n$ is odd and $a_n$ is positive, the graph falls to the left and rises to the right.
  • If $n$ is odd and $a_n$ is negative, the graph rises to the left and falls to the right.
  • If $n$ is even and $a_n$ is positive, the graph rises to the left and right.
  • If $n$ is even and $a_n$ is negative, the graph falls to the left and right.
  • The leading coefficient $a_n$ and the degree $n$ are crucial in determining the end behavior.

Mathematical Notation

$a_n$ is the leading coefficient of the polynomial$n$ is the degree of the polynomial↑ represents rising behavior↓ represents falling behaviorRemember to use proper notation when solving problems

Why It Works

The end behavior of polynomial functions is determined by the highest degree term, $a_n x^n$, as it dominates the behavior as $x$ approaches infinity.

Remember

For polynomials, focus on the leading term $a_n x^n$ to predict end behavior.

Quick Reference

End Behavior (Odd, Positive):Graph falls left, rises rightEnd Behavior (Odd, Negative):Graph rises left, falls rightEnd Behavior (Even, Positive):Graph rises both sidesEnd Behavior (Even, Negative):Graph falls both sides

Understanding Leading Coefficient Test

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Beginner Explanation

The end behavior of a polynomial depends on its leading term $a_n x^n$. If the degree $n$ is even, both ends of the graph go the same direction; if $n$ is odd, the ends go in opposite directions. The sign of $a_n$ tells you if the graph rises (positive) or falls (negative).Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is the end behavior of the polynomial $f(x) = -x^3 + 5x$?

A$falls \, left, \, rises \, right$B$rises \, left, \, falls \, right$C$rises \, both \, sides$D$falls \, both \, sides$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Imagine you are designing a roller coaster track. Use the polynomial $h(x) = 2x^4 - 3x^3 + x - 5$ to predict the track's end behavior.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Analyze the polynomial $g(x) = -4x^5 + 2x^3 - x + 7$.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Determine the end behavior of $f(x) = x^4 - 2x^2 + 3$.

A$rises \, both \, sides$B$falls \, both \, sides$C$falls \, left, \, rises \, right$D$rises \, left, \, falls \, right$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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