Comparing Functions Across Representations

Lesson 1 of 2: Extraction and Intercepts

In this lesson:

• Extract properties from any function representation
• Compare intercepts across different forms
• Justify comparisons with specific evidence

What You Will Learn Today

1. Extract properties from algebraic, graphical, tabular, and verbal forms
2. Compare the same property across two functions
3. Read key features from graphs and tables accurately
4. Justify comparisons using evidence from each representation
5. Communicate comparisons referencing both functions

You Already Know This — Sort Of

You've interpreted functions in every form:

• Graphs: read intercepts, peaks, increasing/decreasing regions
• Tables: find output values, spot max/min, estimate rates
• Equations: evaluate, factor, use vertex formula
• Verbal: translate words into numbers or formulas

Today's new challenge: extract the same property from two functions in different forms.

How Each Representation Reveals Properties

Which properties are easiest to see in each form?

Reference: Find Any Property from Any Form

The Same Function, Four Ways

Every form gives the same y-intercept, zeros, and maximum — just found differently.

Example: Extract Maximum from an Equation

Function A (algebraic):

Step 1: Complete the square (or use vertex formula)

Step 2: Read the vertex

Maximum value = 4, occurring at

Example: Extract Maximum from a Graph

Function B (graph): a downward parabola with vertex near

Reading from the graph:

• Peak location: approximately
• Peak height: approximately

Maximum ≈ 4 at

Note the ≈ symbol: graph readings are estimates, not exact values.

Check-In: What Are You Comparing?

Before extracting any property, ask yourself two questions:

Question 1: What property am I comparing?
(maximum? y-intercept? rate of change? zeros?)

Question 2: How do I find that property in each representation?
(compute? read? scan? translate?)

Think: what's the first thing you do before comparing two functions?

Two Steps for Every Comparison

The cross-representation comparison always follows the same structure:

Step 1: Identify the target property

Step 2: Extract that property from each representation using the appropriate technique

Step 3: Compare the extracted values and state your conclusion

This three-step structure works regardless of which two representation types you're comparing.

From Extraction to Comparing Intercepts

We've built the extraction framework.

Now we focus on one specific property: intercepts.

• y-intercepts: where does the function cross the y-axis?
• x-intercepts: where does the function cross (or touch) the x-axis?

Intercepts answer real questions: "Which starts higher?" "Which reaches zero first?"

Finding y-Intercepts Across All Forms

y-intercept = the function's value when

• Algebraic: evaluate
• Graph: read where the curve crosses the y-axis
• Table: find the row where input = 0
• Verbal: translate first, then evaluate at zero

Example: Compare y-Intercepts (Table vs. Algebra)

A (table): row shows

B (algebraic): , so

Comparison:

"Function B has the greater y-intercept."

Finding x-Intercepts Across All Forms

x-intercept = input where output equals zero

• Algebraic: solve
• Graph: read x-axis crossings
• Table: find rows where output = 0

A table shows only listed values — zeros may fall between entries or not appear at all.

Example: Compare x-Intercepts (Table vs. Algebra)

A (table): zeros at and

B (algebraic): , so or

"Both have two x-intercepts — at different locations."

Guided Practice: Graph vs. Algebra Intercepts

A (graph): x-intercepts near and

B (algebraic):

1. Find the exact x-intercepts of Function B.
2. Which function's zeros are closer to the origin?
3. Compare the y-intercepts.

Try each step before the next slide.

Quick Check: y-Intercept from Words

Verbal description: "A car's value starts at $28{,}000 and decreases by $3{,}500 each year."

What is the y-intercept of the function , where is years?

Think: what does equal?

Real Context: Which Project Breaks Even Sooner?

• Project A: zero crossing at months
• Project B: zero crossing at months

"Project A breaks even sooner — x-intercept at , two months before Project B."

Your Turn: Two Intercept Comparisons

Problem 1: A (table): , , . B (algebraic): .
Compare the y-intercept and x-intercept of each.

Problem 2: A (graph): y-intercept ; zeros at and . B: .
Write two comparison statements.

Pause and try before the next slide.

Intercept Comparison Practice: Answers Revealed

Problem 1:

• y-intercepts: vs. . A is greater.
• x-intercepts: A at ; B at . A is smaller.

Problem 2:

• y-intercepts: A: ; B: . A is greater.
• x-intercepts: B factors as — same zeros. Equal.

Key Takeaways from Lesson One

✓ Identify the target property before extracting
✓ Match technique to the representation
✓ Algebraic gives exact values; graphs give estimates

Graph readings: use ≈, not =
Tables may miss zeros between entries
Compare both functions; use a comparison word

What Is Coming in Lesson Two

Lesson 2 covers:

• Maxima and minima — the standard's signature example
• Rates of change compared across intervals
• Open comparisons: you choose the property
• Full justification writing template