Exponential vs. Linear: Who Wins?

Lesson 1 of 2: Comparing with Tables and Graphs

In this lesson:

• Observe that exponential growth eventually exceeds linear growth
• Compare exponential to quadratic and higher polynomial growth
• Identify crossover points using tables and graphs

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Use tables to observe that an exponential eventually exceeds any linear function
2. Use tables to observe that an exponential eventually exceeds quadratic and higher-degree polynomials
3. Identify the crossover point where exponential overtakes a polynomial

Which of These Two Functions Grows Faster?

Make a prediction before we compute:

• — linear, slope of 1000
• — exponential, doubles each step

At , which is larger?

Think first, then advance to see the table.

Constant Ratio vs. Constant Difference

Linear : adds the same amount each step

• — always

Exponential : multiplies by the same factor each step

• — always

The multiplier is the key difference.

Building a Table: Watching the Crossover

The exponential starts far behind — but notice what happens after .

Crossover: Exponential Takes the Lead

At : vs.

The exponential overtakes the linear between and .

• Before crossover: linear is ahead
• At crossover: they're approximately equal
• After crossover: exponential pulls away permanently

Seeing It on a Graph

• For small : the line is above the curve
• At the crossover: they intersect
• For large : the curve rockets upward

A Steeper Slope Only Delays the Crossover

vs. :

Check Your Understanding: Linear vs. Exponential

At , which is larger: or ?

Compute both and compare.

Answer: Exponential Already Won at

The exponential is larger:

At , the exponential already overtook the linear.

From Linear to Quadratic: Does Acceleration Help?

Quadratic functions grow faster and faster — can they keep up?

• Linear : adds same amount each step
• Quadratic : each step adds more than the last
• Exponential : multiplies each step

Prediction: Does beat for large ?

Comparing Against in a Table

• matches at and
• After : exponential pulls away permanently
• At : vs.

Check Your Understanding: Quadratic vs. Exponential

At , which is larger: or ?

Think before you advance.

Answer: Exponential Is 10 Times Larger Here

The exponential is larger — by a factor of more than 10.

By : , — a factor of over 2,600.

Does the Cubic Function Keep Up?

• Cubic is ahead at : vs.
• Crossover between and
• At : vs.

Higher Degrees — More Delay, Same Result

Every polynomial eventually loses — no matter how high the degree:

• vs. : crossover near
• vs. : crossover is very large but finite

The degree of the polynomial only delays the crossover.

Prediction: Can Beat ?

Could ever be overtaken by ?

• grows extremely slowly at first
• is enormous for moderate values of

What do you predict?

Yes — Every Exponential with Wins

Yes — every exponential with base eventually wins.

A base closer to 1 means the crossover is much, much later.

But the crossover always exists.

What You Observed Today — Key Takeaways

• Exponential eventually exceeds any linear or polynomial function
• Higher degree delays the crossover; it never prevents it

"Eventually" matters — polynomial can be far ahead early on.

Bigger coefficient shifts crossover right; never eliminates it.

Quadratic accelerates but has no constant ratio — not exponential.

Coming Up in Lesson 2

Why does exponential always win?

• Analyzing growth increments: additive vs. multiplicative
• The savings account analogy
• Finding crossover points with tables and graphing technology
• Real-world decision-making: when does exponential become the better deal?