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End Behavior of a Function

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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 approaches

Remember 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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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

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

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

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

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a skateboard ramp represented by $f(x) = x^3 - 3x + 2$. What is the end behavior?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

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

Recap

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