Proving Equal Differences and Equal Factors

Lesson 1 of 2: The Algebraic Proofs

In this lesson:

• Prove linear functions grow by equal differences
• Prove exponential functions grow by equal factors
• Verify both proofs with specific numbers

What You Will Learn Today

1. Prove that produces equal differences over equal intervals
2. Prove that produces equal factors over equal intervals
3. Verify proof formulas using specific numerical examples
4. Explain what distinguishes a proof from checking a few cases

Motivation: Are the Differences Always the Same?

Differences: 13-7 = 6, 19-13 = 6, 25-19 = 6

"OK, the differences are 6 for these inputs. But will they STILL be 6 at ?"

What a Proof Does That a Table Cannot

A table checks specific inputs. A proof covers every real number .

The key: a proof uses a variable , not a number.

When simplifies to a result with no remaining, that result holds for every simultaneously — because dropped out.

Step 1: Compute

— substitute for :

Key: multiplies the entire . The term is new — it will survive after subtraction.

Step 2: Compute

The and terms cancel. Only survives.

No in the result → holds for every . Proof complete.

Why the Starting Point Cancels Out

• appears in both terms → cancels
• appears in both terms → cancels
• Only remains — the shift's contribution

has no . The starting point doesn't matter.

Verify with Numbers: Linear Proof

Proof: for , → difference = .

Check :

Check :

Quick Check: Apply the Linear Proof

For with equal spacing :

1. Compute — substitute carefully
2. Compute — simplify
3. What is the constant difference?

Try each step before the next slide.

From Equal Differences to Equal Factors

Linear functions: constant difference

Now: exponential functions and constant factors

Key tool needed: The exponent rule

This rule separates the "where you start" () from the "how much you grow" ().

Motivation: Are the Ratios Always the Same?

Ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2

"The ratios are always 2 for these inputs. Will they STILL be 2 at ?"

Step 1: Compute

— substitute for in the exponent:

Key: the exponent rule separates "where you start" () from "how you grow" ().

Step 2: Compute

The factors cancel. The factors cancel. Only survives.

This holds for every value of — because is no longer in the result.

Same Cancellation Structure in Exponential Proof

• appears in numerator and denominator → cancels
• appears in numerator and denominator → cancels
• Only remains — same pattern as the linear proof

Verify with Numbers: Exponential Proof

Proof: , → ratio = .

Check :

For : ratio = , so every 3-unit step multiplies by 8.

Quick Check: Apply the Exponential Proof

For with equal spacing :

1. Compute — use the exponent rule
2. Form and simplify
3. What is the constant ratio?

Try before the next slide.

Your Turn: Write Both Proofs

P1: Prove has equal differences. Give the constant difference for .

P2: Prove has equal factors. Give the constant factor for .

Show all algebra. Pause before the next slide.

Answers: Linear and Exponential Proofs

P1: ✓

P2: ✓

Key Takeaways from Lesson One

✓ Linear: — cancels by subtraction
✓ Exponential: — cancels by division
✓ Starting point drops out — that makes it a proof

One numerical check is verification, not proof
Distribute over all of
Use , not addition

Coming Up in Lesson Two

Deck 2 covers:

• What reveals about slope
• What reveals about the base
• Using the proofs to classify any algebraic function
• Testing functions that are neither linear nor exponential