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Nonlinear Functions

Read parabolas and exponential curves the way the SAT asks: vertices, zeros, growth factors, and what each form reveals instantly.

⏱ 9 min📍 Learn first: Nonlinear Equations and Systems
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Learn Nonlinear Functions in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:48

Master the Seven Non-Linear Function Families

A short spoken refresher on this concept.

00:00 / 23:55

Seven mathematical shapes that govern reality

The full lesson, worked through in depth.

Concept Snapshot

Definition
Functions whose rate of change varies: quadratics (parabolas) and exponentials (growth/decay curves) dominate the SAT.
What the SAT tests
Vertex and zeros from different quadratic forms; exponential growth/decay models like f(t) = a(1 + r)ᵗ.
Key rule
Vertex form a(x − h)² + k shows the vertex (h, k). Factored form shows zeros. f(t) = a·bᵗ: a = start, b = growth factor.
Most common mistake
Writing 3% growth as b = 0.03 instead of b = 1.03.
Prerequisite
Nonlinear Equations and Systems

Visual Explanation

What to picture: Vertex form a(x − h)² + k shows the vertex (h, k). Factored form shows zeros. f(t) = a·bᵗ: a = start, b = growth factor.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionFunctions whose rate of change varies: quadratics (parabolas) and exponentials (growth/decay curves) dominate the SAT.
Key ruleVertex form a(x − h)² + k shows the vertex (h, k). Factored form shows zeros. f(t) = a·bᵗ: a = start, b = growth factor.

Nonlinear Functions

Advanced Math

Form → FeatureEach form reveals one feature instantly — match the question to the form.
Exponential ModelingBuild f(t) = a·bᵗ from a starting value and a percent change.
Linear vs. ExponentialConstant difference → linear; constant ratio → exponential.

How the SAT Tests It

Pattern

Form → Feature

Each form reveals one feature instantly — match the question to the form.

Clue: "vertex", "x-intercepts", "minimum value"
Pattern

Exponential Modeling

Build f(t) = a·bᵗ from a starting value and a percent change.

Clue: "increases by 4% each year"
Pattern

Linear vs. Exponential

Constant difference → linear; constant ratio → exponential.

Clue: A table asking "which model…"

Worked Example

A ball's height is h(t) = −16(t − 2)² + 70. What is its maximum height, and when does it occur?

Maximum of a downward parabola — that's the vertex.
Vertex form: a(t − h)² + k with a = −16, h = 2, k = 70. No algebra needed.
Vertex (2, 70): maximum height 70 feet at t = 2 seconds.
a is negative → opens downward → the vertex is a maximum, not minimum. ✓

Try It Yourself

A population of 12,000 bacteria decreases by 8% every hour. Which function models the population after t hours?

Common Traps

What students do

Using the percent itself as the base.

Why it fails

bᵗ multiplies repeatedly — a base of 0.03 would erase 97% of the quantity every step instead of growing it 3%.

Do this instead

Base = 1 + r for growth, 1 − r for decay. Sanity-check: after one step, does f(1) move the right direction by the right amount?

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Seven mathematical shapes that govern reality

Which form answers which question, plus growth-factor fluency.

00:00 / 23:55

23:55 · the full lesson

What to notice

Vertex form a(x − h)² + k shows the vertex (h, k). Factored form shows zeros. f(t) = a·bᵗ: a = start, b = growth factor.

What the lesson covers

  • 0:00 Three quadratic forms
  • 1:50 Vertex reading
  • 3:20 Exponential bases
  • 5:00 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Nonlinear Functions notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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