Comparing Two Functions

Lesson 2 of 3: Functions (Deck 2 of 2)

In this lesson:

• Compare two functions in different forms
• Decide which is better — and on what terms

Learning Objectives: Comparing Function Representations

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

1. Compare the rate of change of two functions in different representations
2. Compare the initial values of two functions in different representations
3. Determine which function is greater at a specific input

Which Phone Plan Is Cheaper?

Plan A (a friend tells you): $20 a month plus $0.05 per text.

Plan B (an ad shows a table):

One's in words, one's a table. Now what?

Pull the Two Numbers from Each

Plan A (words): initial $20, rate $0.05/text.

Plan B (table): initial $30, rate /text.

Now they're comparable.

The Four-Step Comparison Protocol to Use

1. Identify the property the question asks about
2. Extract it from Function A
3. Extract the same property from Function B
4. Compare and state the conclusion

Identify first — so you don't compare a rate to a starting value.

Example 1: Table vs. Equation

A (table): — rate .

B: — rate .

Example 2: Graph vs. Words

A (graph): crosses at . B (words): $20 fee. B starts higher.

Example 3: Table vs. Graph at

A (table) at : . B: , so . B is greater.

Quick Check: Which Has Greater Rate?

A (table):

B:

Which has the greater rate of change? Mind the -step.

Predict: Is the Steeper One Always Bigger?

A: (rate , starts at )

B: (rate , starts at )

At , which is bigger? Predict before advancing.

Steeper Is Not Always Bigger

At : .

B is bigger — even though A grows faster.

A catches up later: at .

"Which Is Better" Depends on the Question

For the phone plans, three different questions, three possible answers:

• Lower base cost? Plan A ($20 vs $30)
• Lower per-text rate? Plan B ($0.03 vs $0.05)
• Cheaper at 400 texts? Compute and see

Phone Plans at 400 Texts

Plan A is cheaper at 400 texts.

Guided: Two Runners at 12 Seconds

A (graph): through and rate , so .

B (words): starts m ahead, runs m/s .

Who's farther at ? Set it up, then advance.

Find the Error in This Comparison

A student compares A (rate , from a table) with B: and writes:

", so B is greater."

What did they mix up?

Your Turn: Decide on Your Own

On your whiteboard — greater rate of change?

A (table): B:

Write the fraction for A. No calling out.

Make the Call: Two Savings Accounts

A: . B (table): .

Answer all three, then recommend:

1. Which started with more?
2. Which grows faster per week?
3. Who has more at week 10?

Comparing Functions: The Key Takeaways

✓ Extract, label, compare — never need the same format

✓ Steeper isn't always bigger — compute at the input

✓ "Better" depends on the question: start, rate, or value at a point

Next: build the functions yourself — write from a situation.