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Graphing Polynomial Functions

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Graphing Polynomial Functions

Study Guide

Key Definition

A polynomial function is an expression of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $n$ is a non-negative integer.

Important Notes

The degree of a polynomial is the highest power of $x$.
Even degree polynomials have graphs with similar end behaviors.
Odd degree polynomials have graphs with opposite end behaviors.
The leading coefficient affects the direction of the graph's ends.
Intercepts and turning points are key features of polynomial graphs.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$x^n$ represents $x$ raised to the power of $n$

Remember to use proper notation when solving problems

Why It Works

Polynomial functions are smooth and continuous, allowing us to predict and understand their behavior using $algebraic$ methods.

Remember

Polynomial graphs change direction at turning points, which can be found by setting the derivative to zero.

Quick Reference

Degree 0:Constant function: $f(x) = a$

Degree 1:Linear function: $f(x) = ax + b$

Degree 2:Quadratic function: $f(x) = ax^2 + bx + c$

Understanding Graphing Polynomial Functions

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $f(x) = ax + b$ for linear functions.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which graph represents $f(x) = x^2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is shaped like a parabola. If the equation is $f(x) = -0.5x^2 + 3x$, find the maximum height.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

Identify the roots of $f(x) = x^3 - 6x^2 + 11x - 6$.

Recap

Watch & Learn

Review key concepts and takeaways