Slope, Base, and Classifying Functions Algebraically

Lesson 2 of 2: Connections and Classification

In this lesson:

• Connect proof results to slope and base
• Compute growth factors over different intervals
• Classify functions using algebraic tests

What You Will Learn Today

1. Connect slope to the constant difference per unit interval
2. Connect the base to the constant ratio per unit interval
3. Compute growth factors for intervals of any width
4. Apply the difference and ratio tests to classify functions
5. Identify functions that are neither linear nor exponential

You Have the Proof — Now Use It

From Lesson 1:

• Linear:
• Exponential:

These formulas do more than prove a pattern. They tell us what slope and base actually mean — and they give us tests to classify functions we've never seen before.

Linear Connection: Proof Reveals Slope

From the proof:

Slope IS the constant rate of change — the proof shows this holds for every , not just pairs in a table.

Example: Reading Slope from the Proof Formula

, spacing :

Rate of change:

The slope is 6 — which we could read directly from the formula, but the proof tells us WHY it's constant.

Exponential Connection: Proof Reveals Base

From the proof:

• : ratio = (the base itself)
• : ratio =
• : ratio =

Base = ratio per unit; for interval , the ratio is .

Example: Growth Factors at Different Interval Widths

— 6% annual growth

• : factor = (6% growth)
• : factor = (33.8% growth)
• : factor = (79.1% growth)

Quick Check: Compute a Growth Factor

— a savings account growing at 3% per year.

What is the growth factor over 20 years?

Use the proof formula. Think before the next slide.

Using Proofs to Classify: The Strategy

Linear: is constant (no in the result).

Exponential: is constant (no in the result).

Both fail: neither linear nor exponential.

Example: Classify a Quadratic Function Algebraically

Difference: — depends on → not linear.

Ratio: — depends on → not exponential.

Conclusion: quadratic — neither type.

Example: Testing a Shifted Exponential

Ratio test:

This depends on → NOT exponential.

Only pure passes the ratio test.

Quick Check: Is This Exponential?

Apply the ratio test: compute and simplify.

Does the result depend on ?

Think before the next slide.

Summary: Classifying Four Functions Side by Side

Guided Practice: Classify a Function

Difference test:

This depends on → not linear. Now try the ratio test.

— complete this step yourself.

Your Turn: Classify These Three Functions

For each: apply the difference test (and ratio test if needed). State the type and cite evidence.

Pause before the next slide.

Classification Practice: Answers and Evidence

1. → Linear ✓

2. → Exponential ✓

3. ; ratio also has → Neither ✓

Key Takeaways from Lesson Two

✓ Slope = — proof shows it's constant for every
✓ Base = ratio per unit interval ()
✓ Growth factor over units =

Use differences for linear; ratios for exponential
fails the ratio test — add nothing to a pure exponential
Result depends on ? Test fails — no exceptions

Connecting These Proofs to Future Modeling

You've completed HSF.LE.A.1.a.

Coming up:

• HSF.LE.A.1.b: Recognize constant-rate situations
• HSF.LE.A.1.c: Recognize constant percent-rate situations
• HSF.LE.A.2: Construct models from data