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Comparing Linear, Polynomial, and Exponential Growth

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Comparing Linear, Polynomial, and Exponential Growth

Study Guide

Key Definition

Linear growth is represented by a straight line, polynomial growth by curves whose shape depends on the degree of the polynomial, and exponential growth by an upward curve. Their general forms are $y = m x + b$, $y = a x^n$, and $y = a b^x$, respectively.

Important Notes

Linear growth is represented by functions like $y = x$.
Polynomial growth is represented by functions like $y = a x^n$ (e.g., $y = x^4$).
Exponential growth is represented by functions like $y = 2^x$.
Exponential growth overtakes polynomial growth over time.

Mathematical Notation

$y = x$ represents a linear function.

$y = x^4$ represents a polynomial function.

$y = 2^x$ represents an exponential function.

Remember to use proper notation when writing functions. General forms: linear $y = m x + b$; polynomial $y = a x^n$; exponential $y = a b^x$.

Why It Works

Each type of growth is represented by a different function. These functions, when graphed, show different rates of increase. This is why the shapes of the graphs differ.

Remember

Exponential growth, represented by $y = 2^x$, will eventually overtake polynomial growth, represented by $y = x^4$, as $x$ increases.

Quick Reference

Linear Growth:$y = x$

Polynomial Growth:$y = x^4$

Exponential Growth:$y = 2^x$

Understanding Comparing Linear, Polynomial, and Exponential Growth

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Linear growth refers to an increase at a constant rate, generally of the form $y = m x + b$, represented by a straight line when graphed.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which function represents exponential growth?

Real-World Problem

Question Exercise

Intermediate

Population Growth Scenario

In a certain town, the population has been growing exponentially. If the population is represented by $y = 2^x$, where $x$ is the number of years, explain why this population grows faster compared to a town with linear growth represented by $y = x$?

Advanced Challenge

Thinking Exercise

Advanced

Think About This

Given the functions $y = x$, $y = x^4$, and $y = 2^x$, at what point does exponential growth overtake polynomial growth?

Recap

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Review key concepts and takeaways