Comparing Extrema, Rates, and Open Analysis

Lesson 2 of 2: Maxima, Rates, and Justification

In this lesson:

• Compare maxima and minima across representations
• Compare average rates of change over intervals
• Choose properties to compare and write full justifications

What You Will Learn Today

1. Compare maxima and minima from graphs and equations
2. Compare average rates of change across representations
3. Recognize when graph estimates are sufficiently precise
4. Choose which properties to compare in open tasks
5. Justify comparisons with evidence from both functions
6. Write full comparison sentences citing both functions

Hook: Which Peak Is Higher?

The standard's own example asks this exact question:

• Function A (graph): parabola with vertex near
• Function B (algebraic):

"Which has the larger maximum?"

How would you answer this? Think before the next slide.

Comparing Maxima and Minima Across Forms

Key difference: algebraic gives exact values; graph gives estimates.

Example: Maximum Comparison (Graph vs. Algebra)

Function A (graph): maximum ≈ 5 at

Function B (algebraic): → maximum = 5 at

"Both functions have the same maximum value of 5."

Worked: Complete the Square to Find Maximum

Step 1: Factor out

Step 2: Complete the square

Step 3: Read the vertex

Precision: When Does Graph Reading Work?

Graph readings are approximate. Algebraic values are exact.

• Values differ clearly (≈ 5 vs. exact 8): graph reading is sufficient
• Values are close (≈ 4.9 vs. exact 5): note the uncertainty

Rule: Always use ≈ for graph readings.

Quick Check: Estimate vs. Exact

A student reads a graph and says: "The maximum of Function A is 6."

Then they compute: "The maximum of Function B is ."

They conclude: "Function A has the larger maximum."

What's wrong with this conclusion?

Think before the next slide.

Comparing Rates of Change Across Forms

Average rate of change over :

Example: Rate Comparison (Verbal vs. Table)

A (verbal): Car A goes from 0 to 60 mph in 8 s.

B (table): speed at is 0; at is 50 mph.

"Car A has the greater average acceleration."

Guided Practice: Compare Rates of Change

A (graph): line through and

B (algebraic):

Compare the average rate of change over :

1. Find Rate A from the graph coordinates.
2. Find Rate B using the formula.
3. Write a comparison sentence.

Try each step before the next slide.

Quick Check: Fair Comparison of Rates

Statement: "Function A increases by 12 from to . Function B increases by 20 from to . Function B has a greater rate."

Is this a valid rate comparison? Why or why not?

Think before the next slide.

From Targeted to Open Comparisons

So far: the question specifies which property to compare.

Open comparisons: you choose the property.

Good open comparisons are:

1. Specific — name the exact property
2. Supported — cite values from both functions
3. Complete — use a comparison word

Example: Open Comparison (Linear vs. Exponential)

1. y-intercepts: A = 2, B = 1. A starts higher.
2. Rate: A is constant; B's grows. B eventually changes faster.
3. Long-run: Exponential B will overtake linear A.

Writing a Complete Comparison Sentence

Template: "Function A has a [larger/smaller/equal] ___ than B because [value A] [>/</=] [value B]."

Examples:

• "A has a larger y-intercept than B: while ."
• "Both maxima equal 5: A's vertex is at and B's algebraic maximum equals 5."

Guided Practice: Choose and Compare Properties

Function A (graph): a decreasing line with y-intercept at 6 and x-intercept at 4

Function B (algebraic):

Write three comparison statements, covering different properties.

Think: what can you easily extract from each representation?

Quick Check: Is This a Valid Comparison?

Student's answer: "Function A has a maximum of 5. Function B increases on ."

What's wrong? How would you fix it?

Think before the next slide.

Your Turn: Write Open Comparison Statements

P1: A (table): . B: .
Write two comparisons.

P2: A (algebraic): . B (graph): vertex at , opens upward.
Write two comparisons.

Pause and try before the next slide.

Open Comparison Practice: Answers Revealed

P1:

• Rates: A = 2 (constant); B increases. B's rate eventually exceeds A's.
• y-intercepts: vs. . A is greater.

P2:

• A has a maximum; B has a minimum. Different extreme types.
• A: max = 4; B: min ≈ 2. A has the larger extreme.

Key Takeaways from Lesson Two

✓ Use ≈ for graph readings; = for algebraic
✓ Rate comparisons: same interval required
✓ Open comparisons: specific, supported, complete

Near-equal values from graph vs. algebra — note it
does not mean A is always greater
Describing ≠ comparing; use a comparison word

Connecting to Future Function Topics

You've completed the HSF.IF domain.

This skill connects forward to:

• HSF.LE: Compare linear and exponential models
• HSF.BF: Build functions matching given properties
• Modeling: Compare a formula to observed data