Unit 12 · Polynomials, Modeling & Comparison

Comparing Linear, Exponential, and Quadratic Models

Linear: first differences constant. Exponential: ratios constant. Quadratic: second differences constant. Context also matters: constant amount vs constant percent vs symmetric vertex stories.

Algebra I Regents

AI-F.LE.1AI-F.IF.4AI-F.IF.9

Notes

Look at how it grows

A table 3, 6, 9, 12 is linear (+3). A table 3, 6, 12, 24 is exponential (×2). A table 3, 6, 11, 18 has first differences 3, 5, 7 (second differences 2) — quadratic.

Vocabulary

first difference

The change from one y-value to the next in a table with uniform x-steps.

Example: For 2, 5, 8, 11 the first differences are all 3 (linear).

Worked examples

Try the prompt first. The steps stay folded until you ask to see them.

Choose a model

A rumor spreads: 4, 12, 36, 108 people after 0, 1, 2, 3 hours. Linear, exponential, or quadratic?

Common errors

Watch for

Calling any curve exponential

Parabolas curve too, but they add a changing amount, they do not multiply by a constant.

Fix: Compute ratios and second differences.

Explore with an applet

Guided practice

Submit each answer to check it. The key is not shown until you do.

Question 1 of 3Algebra I RegentsAI-F.LE.3Algebra I Regents · June 2023 · Q7Non-calculatorproficient

Which function will have the greatest value when x > 1?

g(x) = 2(5)ˣf(x) = 2x + 5h(x) = 2x² + 5k(x) = 2x³ + 5

Choose or type an answer first. The key stays hidden until you do.

Question 2 of 3Algebra I RegentsAI-F.LE.3Algebra I Regents · June 2023 · Q7Non-calculatorproficient

Which function will have the greatest value when x > 1?

g(x) = 2(5)ˣf(x) = 2x + 5h(x) = 2x² + 5k(x) = 2x³ + 5

Choose or type an answer first. The key stays hidden until you do.

Question 3 of 3Algebra I RegentsAI-F.LE.3Algebra I Regents · June 2023 · Q7Non-calculatorproficient

Which function will have the greatest value when x > 1?

g(x) = 2(5)ˣf(x) = 2x + 5h(x) = 2x² + 5k(x) = 2x³ + 5

Choose or type an answer first. The key stays hidden until you do.

Intervention

Color-code three tables and compute differences/ratios as a routine.

Enrichment

Over a long interval, exponential growth outruns quadratic, which outruns linear (for typical positive coefficients).

Explain your reasoning

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I chose this family because the table/context shows…