Graphing Polynomial Functions - Varsity Tutors

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Graphing Polynomial Functions

Study Guide

Key Definition

A polynomial function is expressed as $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where each term is of the form $a_ix^i$.

Important Notes

  • A polynomial of degree zero is a constant function: $f(x) = a$
  • A polynomial of degree one is a linear function: $f(x) = ax + b$
  • A polynomial of degree two is a quadratic function: $f(x) = ax^2 + bx + c$
  • Graph shapes differ based on whether the highest degree is even or odd
  • The leading coefficient affects the graph's direction

Mathematical Notation

$x^n$ represents x raised to the power of n$+$ represents addition$-$ represents subtraction$\times$ represents multiplication$\div$ represents divisionRemember to use proper notation when solving problems

Why It Works

Understanding polynomial functions helps in predicting the shape and behavior of their graphs based on their degree and coefficients.

Remember

The highest degree of the polynomial determines the shape and complexity of the graph.

Quick Reference

Linear Function:$f(x) = ax + b$Quadratic Function:$f(x) = ax^2 + bx + c$

Understanding Graphing Polynomial Functions

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

Simple explanation with $y = ax + b$, a linear function.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is the graph of $f(x) = 3x + 2$?

A$\text{Line with positive slope}$B$\text{Horizontal line}$C$\text{Parabola}$D$\text{Circle}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

A skateboarder jumps off a ramp. The height $h$ of the jump can be modeled by $h(t) = -4.9t^2 + 5t + 1$, where $t$ is time.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Consider the polynomial $f(x) = x^3 - 6x^2 + 11x - 6$. Determine the roots, end behavior, and relative extrema of the graph.

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

Challenge Quiz

Single Choice QuizAdvanced

Which polynomial function has a graph that opens downward?

A$f(x) = -x^2 + 4x - 4$B$f(x) = x^2 + 4x + 4$C$f(x) = 2x^3 - 3x + 1$D$f(x) = x^4 - 5x^2 + 6$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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