Why Exponential Always Wins

Lesson 2 of 2: Mechanism and Crossover Points

In this lesson:

• Explain why exponential growth dominates using increments
• Locate crossover points using tables and graphs
• Apply the dominance principle to real-world decisions

What You Will Learn Today

By the end of this lesson, you can:

4. Explain why exponential dominates: multiplicative growth compounds; additive growth does not
5. Interpret graphs of exponential and polynomial functions together, identifying which region each leads
6. Apply the dominance principle to real-world contexts

Recall: What Did Lesson 1 Show?

From Lesson 1, you observed:

• eventually exceeds
• eventually exceeds
• Larger coefficients and higher degrees only delay the crossover

Today's question: Why does this always happen?

The Key Difference: Additive vs. Multiplicative

Linear : increment is always

Quadratic : increment grows, but how fast?

Exponential : increment multiplies each step

The key: what happens to the increment as grows large?

Linear Increments Are Always Constant

For , the step-by-step increment is:

• At : adds
• At : still adds

The linear function adds the same amount forever.

Quadratic Increments Grow Linearly, Not Exponentially

For , the step-by-step increment is:

• At : adds
• At : adds

The increment grows — but only linearly.

Exponential Increments: They Equal the Current Value

For : each increment equals its current value.

Savings Account: Additive vs. Multiplicative

Account A: Deposit 1,000/year (additive)

Account B: Earn 10% on balance (multiplicative)

Once Account B's balance exceeds 10,000, interest beats the flat deposit.

Check: What Is 's Increment at ?

At :

The increment equals the current value.

Think: at , what will the increment be?

Answer: At the Increment Exceeds One Million

Meanwhile: still adds per step.

The exponential's increment has grown a thousandfold.

From Mechanism to Finding Crossovers

You know why exponential wins.

Now: how do you find when it happens?

Two methods:

1. Build a table and narrow down the crossover
2. Use a graph to visualize the intersection

Visualizing the Crossover on a Graph

For : line is above curve. For : curve rockets up.

Narrowing Down: Table Zoom Near

At : — crossover between 26 and 27.

Real-World Crossover: Two Monthly Plans

Plan A: Starts at $100/month, grows ×1.5 each month (exponential)

Plan B: Pays $100/month, adds $100 each month (linear)

Question: When does Plan A become the better deal?

Build a table for months 1 through 30.

Plan A vs. Plan B: Building the Table

Plan A is far ahead by month 5.

Crossover Decisions Depend on Time Horizon

Before crossover: the polynomial option may be better

After crossover: exponential option is permanently better

Real-world lesson: the same choice can be wrong or right depending on your time horizon.

Small Bases Win Too — Just Later

• vs. : crossover near
• vs. : crossover near
• vs. : crossover near

A base closer to 1 just requires a longer wait — but the crossover always exists.

Could Ever Beat ?

At : vs.

At : vs.

The crossover is near — very large, but finite.

Answer: Yes — Crossover Near

Yes — overtakes near .

Base closer to 1 means a later crossover. But it always exists for any .

This is the general principle from HSF.LE.A.3.

Key Takeaways from Both Lessons

• Exponential increment equals current value — it compounds
• Every () eventually exceeds every polynomial
• Crossover always exists; small bases just shift it further right

Polynomial wins early; exponential wins long-term.

Quadratic accelerates but its increment is linear, not exponential.

Coming Up: Finding Crossovers with Logarithms

In the next lesson (HSF.LE.A.4):

• Use logarithms to solve directly
• Find exact crossover -values without tables
• Apply to compound interest and growth problems

You already understand why it works — now you'll have the tools to find the exact answer.