End Behavior Of A Function - Varsity Tutors

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End Behavior of a Function

Study Guide

Key Definition

The end behavior of a polynomial function is the behavior of the graph of $f(x)$ as $x$ approaches $+\infty$ or $-\infty$.

Important Notes

  • The degree of the polynomial function determines if it is even or odd.
  • The sign of the leading coefficient affects the direction of the end behavior.
  • For an even degree with a positive leading coefficient, $f(x) \to +\infty$ as $x \to \pm\infty$.
  • For an even degree with a negative leading coefficient, $f(x) \to -\infty$ as $x \to \pm\infty$.
  • For an odd degree with a positive leading coefficient, $f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to +\infty$ as $x \to +\infty$; with a negative leading coefficient, the directions are reversed.

Mathematical Notation

$\to$ means approachesRemember to use proper notation when solving problems

Why It Works

The degree and leading coefficient of the polynomial determine the direction of the graph's tails, which is crucial for understanding end behavior.

Remember

Even-degree polynomials with positive leading coefficients rise to $+\infty$ on both ends.

Quick Reference

Even Positive:$f(x) \to +\infty$ as $x \to \pm\infty$Even Negative:$f(x) \to -\infty$ as $x \to \pm\infty$Odd Positive:$f(x) \to -\infty$ as $x \to -\infty$; $f(x) \to +\infty$ as $x \to +\infty$Odd Negative:$f(x) \to +\infty$ as $x \to -\infty$; $f(x) \to -\infty$ as $x \to +\infty$

Understanding End Behavior of a Function

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Beginner Explanation

For an even degree and positive leading coefficient, the graph rises to $+\infty$ on both ends.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What is the end behavior of $x^2$?

A$f(x) \to +\infty$ as $x \to \pm\infty$B$f(x) \to -\infty$ as $x \to \pm\infty$C$f(x) \to -\infty$ as $x \to -\infty$; $f(x) \to +\infty$ as $x \to +\infty$D$f(x) \to +\infty$ as $x \to -\infty$; $f(x) \to -\infty$ as $x \to +\infty$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 a skateboard ramp represented by $f(x) = x^3 - 3x + 2$. What is the end behavior?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Analyze the function $x^4 - 4x^3 + 3x + 25$ and determine its end behavior.

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

Challenge Quiz

Single Choice QuizAdvanced

Determine the end behavior of $-x^3 + 2x^2 - x + 5$.

A$f(x) \to +\infty$ as $x \to -\infty$; $f(x) \to -\infty$ as $x \to +\infty$B$f(x) \to -\infty$ as $x \to -\infty$; $f(x) \to +\infty$ as $x \to +\infty$C$f(x) \to +\infty$ as $x \to \pm\infty$D$f(x) \to -\infty$ as $x \to \pm\infty$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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