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Exponential & Log Functions: Explosive Growth

Students explore exponential functions — the mathematics of explosive growth and rapid decay, from populations to radioactive half-life — and meet the logarithm, the inverse that undoes an exponential.

Grade 11Functions50 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
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Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Graph exponential functions.
  • ✓Tell growth from decay.
  • ✓Explain the base and starting value.
  • ✓Understand logarithms as inverses.
Essential Question

Money in a bank, a viral video, a decaying isotope — all follow the same explosive curve. What is exponential growth, and how do logarithms reverse it?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

The Exponential Grapher

Project this and change a and b in y = a·bˣ. Watch the curve explode upward (growth) or fall toward zero (decay).

📈 Change a and b — growth or decay?Try it
a = 1
b = 2

The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — The Explosive Curve

5 min

Bank interest, viral videos, decay — what curve do they share?

👩‍🏫 Teacher Moves

  • Show an exponential curve.
  • Ask what it models.
  • Set the goal: exponentials.

🎒 Student Actions

  • See the curve.
  • Name examples.
  • Predict the rule.
2

I Do — y = a·bˣ

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Explain a (start) and b (factor).
  • Graph growth (b > 1).
  • Graph decay (b < 1).

🎒 Student Actions

  • Learn a and b.
  • See growth.
  • See decay.
3

We Do — Explore Together

13 min

Class uses the Exponential Grapher.

👩‍🏫 Teacher Moves

  • Change a and b.
  • Classify growth vs. decay.
  • Introduce the logarithm as inverse.

🎒 Student Actions

  • Set a and b.
  • Classify it.
  • Meet the log.
4

You Do — On Your Own

12 min

Students work with exponentials.

👩‍🏫 Teacher Moves

  • Graph exponential functions.
  • Model a real situation.
  • Solve a simple log.

🎒 Student Actions

  • Graph them.
  • Model one.
  • Solve a log.
5

Close — Close

5 min

One function.

👩‍🏫 Teacher Moves

  • Give an exponential to classify.
  • Graph it.
  • Hand out the exit ticket.

🎒 Student Actions

  • Classify it.
  • Graph it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School: Algebra II).

CCSS
F-LE.1

Distinguish between situations modeled by linear and exponential functions.

CCSS
F-IF.7e

Graph exponential and logarithmic functions.

CCSS
F-LE.2

Construct exponential functions from a description or table.

CCSS
F-BF.5

Understand the inverse relationship between exponents and logarithms.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Growth vs. decay picture cards.
  • Sentence frame: “If b > 1, it ___.”
  • Use a doubling table.

Support & Access

IEP / 504
  • Start with doubling (b = 2).
  • Use a table of values.
  • Focus on growth first.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Model compound interest.
  • Solve exponential equations with logs.
  • Explore half-life decay.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻The Exponential Grapher
  • 📐Graph paper
  • 🧮Calculators
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

exponential functiona function with the variable in the exponentbasethe repeated factor bgrowthincreasing exponentially (b > 1)decaydecreasing exponentially (b < 1)logarithmthe inverse of an exponentialhalf-lifethe time to decay by halfasymptotea line the curve approachescompound interestinterest on interest
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
In y = a·bˣ, what does it mean if b is greater than 1?
The function shows exponential growth (it increases rapidly).
QUESTION 2
What is a logarithm?
The inverse of an exponential — it answers “the base raised to what power gives this value?”
QUESTION 3
Give one real-world example of exponential decay.
Sample: radioactive half-life, a cooling object, or a depreciating car’s value.

Going deeper? Solve with logs.

Have students use logarithms to solve exponential equations like 2ˣ = 20 (finding x). A printable exponential-log sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Exponential & Logarithmic FunctionsFlip
Card 1
Term
Tap to flip →
Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Exponential & Logarithmic FunctionsQuiz
Score: 0
1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Exponential & Logarithmic FunctionsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: asymptote, base, compound interest, decay, exponential function, growth, half-life, logarithm
  1. a line the curve approaches
  2. a function with the variable in the exponent
  3. the repeated factor b
  4. the time to decay by half
  5. interest on interest
  6. the inverse of an exponential
  7. decreasing exponentially (b < 1)
  8. increasing exponentially (b > 1)

Part B · Show what you learned

  1. In y = a·bˣ, what does it mean if b is greater than 1?
  2. What is a logarithm?
  3. Give one real-world example of exponential decay.
Answer key — Part A: 1) asymptote · 2) exponential function · 3) base · 4) half-life · 5) compound interest · 6) logarithm · 7) decay · 8) growth
Part B: 1) The function shows exponential growth (it increases rapidly). 2) The inverse of an exponential — it answers “the base raised to what power gives this value?” 3) Sample: radioactive half-life, a cooling object, or a depreciating car’s value.