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Comparing Growth:
Linear to Exponential

Discover why exponential growth eventually dominates every polynomial: linear, quadratic, cubic, and beyond.

Algebra 2

What Is a Growth Rate?

A growth rate describes how quickly a function's output increases as the input \( x \) gets larger. Different families of functions grow at fundamentally different speeds.

Linear
Constant rate of change: adds the same amount each step. \( f(x) = mx + b \)
Quadratic
Rate increases steadily: the differences between outputs grow. \( f(x) = ax^2 + \dots \)
Exponential
Rate multiplies each step: outputs grow by a constant factor. \( f(x) = a \cdot b^x \)

Visualizing the Race

x y 0 2 4 6 8 10 100 200 300 400
Linear: \( y = 20x \)
Quadratic: \( y = 3x^2 \)
Exponential: \( y = 2^x \)

Why Does This Matter?

Understanding growth rates is essential for modelling real-world phenomena: population growth, compound interest, viral spread, and resource consumption. Choosing the wrong model leads to wildly inaccurate predictions.

Finance
Compound interest is exponential: a small rate compounds into enormous gains over decades.
Biology
Bacteria doubling every hour: exponential growth quickly overwhelms linear predictions.
Technology
Moore's Law: computing power roughly doubles every two years, an exponential trend.

The Core Principle

This is true no matter the degree of the polynomial: linear (\( n=1 \)), quadratic (\( n=2 \)), cubic (\( n=3 \)), or any \( x^n \). The exponential always wins in the long run.

Multiplicative vs. Additive

1
Linear functions add a constant each step. From \( x \) to \( x+1 \): output increases by \( m \).
2
Quadratic functions add an increasing amount, but the differences grow linearly.
3
Exponential functions multiply by a constant factor \( b \) each step: \( f(x+1) = b \cdot f(x) \).
4
Multiplication compounds: each step's growth builds on the previous growth, creating a snowball effect that no additive process can match.

Degree vs. Base: The Key Distinction

Polynomial \( x^n \)
The variable is the base, raised to a fixed exponent. Higher degree = faster growth among polynomials, but still polynomial.
Exponential \( b^x \)
A fixed base is raised to the variable exponent. The exponent itself grows, making the function's rate of growth accelerate without bound.

The Table Test

\( x \) \( 10x \) \( x^2 \) \( x^3 \) \( 2^x \)
110112
5502512532
101001001,0001,024
202004008,0001,048,576
3030090027,0001,073,741,824

How to Compare Growth Rates

1
Identify the type of each function: is the variable in the base (\( x^n \)) or the exponent (\( b^x \))?
2
Build a table of values for small \( x \) (0 to 10) and large \( x \) (20, 50, 100).
3
Graph both functions and look for the crossover point where the exponential surpasses the polynomial.
4
Examine end behavior: as \( x \to \infty \), the exponential always dominates, regardless of the polynomial's degree or leading coefficient.
5
State your conclusion using the principle: "Eventually, \( b^x \) exceeds \( x^n \) for all \( b > 1 \) and all \( n \)."

Example 1: \( 3x \) vs. \( 2^x \)

0 1 2 3 4 5 6 7 0 15 30 45 60 Crossover
\( y = 3x \)
\( y = 2^x \)

Example 2: \( x^2 \) vs. \( 2^x \)

0 1 2 3 4 5 6 7 8 9 10 250 500 750 1000 Crossover at x=4
\( y = x^2 \)
\( y = 2^x \)

Example 3: \( x^3 \) vs. \( 2^x \)

\( x \) \( x^3 \) \( 2^x \) Which is larger?
112\( 2^x \)
512532\( x^3 \)
101,0001,024\( 2^x \) (just barely)
153,37532,768\( 2^x \)
208,0001,048,576\( 2^x \) dominates

The Ratio Test

A powerful way to see exponential dominance: examine the ratio \( \dfrac{b^x}{x^n} \) as \( x \to \infty \).

Even \( \dfrac{1.001^x}{x^{1000}} \to \infty \). The base just needs to be greater than 1. No polynomial, however large its degree, can keep up.

All Four Together

0 2 4 6 8 10 500 1000 1500 2000
Linear: \( 50x \)
Quadratic: \( 5x^2 \)
Cubic: \( x^3 \)
Exponential: \( 2^x \)

Common Mistakes

✕ Judging by small values
Just because \( x^3 > 2^x \) at \( x = 5 \) doesn't mean it's always larger. You must check large \( x \).
✕ Confusing \( x^n \) with \( n^x \)
\( x^3 \) is polynomial (variable base, fixed exponent). \( 3^x \) is exponential (fixed base, variable exponent). They are fundamentally different.
✕ Thinking a large coefficient saves the polynomial
\( 1{,}000{,}000 \cdot x^2 \) is still eventually beaten by \( 2^x \). Coefficients only delay the crossover.

Summary Comparison

Property Linear Quadratic Polynomial \( x^n \) Exponential \( b^x \)
Form \( mx + b \) \( ax^2 + \cdots \) \( a_n x^n + \cdots \) \( a \cdot b^x \)
Growth pattern Constant addition Increasing addition Increasing addition Constant multiplication
Rate of change Constant Linear Polynomial (degree \( n-1 \)) Exponential
End behavior Slowest Faster Faster still Fastest
Dominates long-term? No No No ✓ Always

Question 1 of 5

Which function will eventually produce the largest output as \( x \to \infty \)?

Question 2 of 5

At \( x = 10 \), which correctly compares \( x^2 \) and \( 2^x \)?

Question 3 of 5

Why does exponential growth eventually exceed polynomial growth?

Question 4 of 5

For the functions \( g(x) = x^3 \) and \( h(x) = 2^x \), the cubic is larger at \( x = 5 \). What can you conclude?

Question 5 of 5

What does the limit \( \displaystyle\lim_{x \to \infty} \frac{2^x}{x^n} = \infty \) tell us?