Mastering Exponential Functions: Growth, Decay, and Half-Life

Algebra 2 Grades High School 13:38 Video
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Lesson Description

Learn to solve real-world problems involving exponential growth, decay, doubling time, and half-life using mathematical models and logarithmic properties.

Video Resource

Exponential Functions Word Problems - Half Life, Doubling, Growth and Decay

Mario's Math Tutoring

Duration: 13:38
Watch on YouTube

Key Concepts

  • Exponential Growth and Decay Models
  • Half-Life and Doubling Time
  • Logarithmic Properties for Solving Exponential Equations

Learning Objectives

  • Students will be able to create exponential models from word problems.
  • Students will be able to solve exponential equations for unknown variables using logarithms.
  • Students will be able to apply exponential models to real-world scenarios involving growth, decay, half-life, and doubling time.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of exponential functions (y = AB^x) and their applications in real-world scenarios. Briefly discuss the concepts of growth, decay, half-life, and doubling time.
  • Video Presentation (15 mins)
    Play the YouTube video 'Exponential Functions Word Problems - Half Life, Doubling, Growth and Decay' by Mario's Math Tutoring. Encourage students to take notes on the formulas and problem-solving techniques presented.
  • Guided Practice (20 mins)
    Work through each example from the video, pausing to explain the steps and answer questions. Emphasize the importance of correctly identifying the initial value, growth/decay factor, and time period. Demonstrate how to apply logarithmic properties to solve for unknown variables.
  • Independent Practice (20 mins)
    Provide students with similar word problems to solve individually or in pairs. Circulate the classroom to offer assistance and monitor progress.
  • Wrap-up and Assessment (10 mins)
    Review the key concepts and problem-solving strategies covered in the lesson. Administer a short quiz to assess student understanding.

Interactive Exercises

  • Population Growth Simulation
    Use an online simulation or spreadsheet software to model population growth under different conditions (e.g., varying growth rates, initial populations). Students can manipulate the parameters and observe the effects on the population size over time.
  • Radioactive Decay Experiment
    Simulate radioactive decay by repeatedly flipping a coin. Heads means the atom decays, tails means it remains. Track the number of 'atoms' remaining after each 'half-life' and compare the results to the theoretical exponential decay model.

Discussion Questions

  • How do you determine whether a situation represents exponential growth or decay?
  • Why are logarithms necessary to solve for time in exponential equations?
  • How can you adapt the exponential growth/decay model to represent half-life and doubling time scenarios?

Skills Developed

  • Problem-Solving
  • Mathematical Modeling
  • Critical Thinking
  • Logarithmic Manipulation

Multiple Choice Questions

Question 1:

A population of bacteria doubles every 4 hours. If the initial population is 50, what is the population after 12 hours?

Correct Answer: 400

Question 2:

The half-life of a radioactive substance is 20 years. If you start with 100 grams, how much will be left after 60 years?

Correct Answer: 12.5 grams

Question 3:

A car depreciates at a rate of 15% per year. If the car originally costs $30,000, what is its value after 5 years?

Correct Answer: $13,314.41

Question 4:

Which of the following equations represents exponential decay?

Correct Answer: y = 5(0.8)^x

Question 5:

If P = 1000(1.05)^t, what does 1.05 represent?

Correct Answer: The growth rate

Question 6:

To solve for the exponent in an exponential equation, which mathematical tool is most useful?

Correct Answer: Logarithms

Question 7:

What does 'A' stand for in the exponential decay formula A = P(1-r)^t?

Correct Answer: The amount after time 't'

Question 8:

In a half-life problem, what value is typically used as the base of the exponential function?

Correct Answer: 1/2

Question 9:

What is the first step in solving for t in the equation 5000 = 200(1.1)^t?

Correct Answer: Divide 5000 by 200

Question 10:

What is the growth rate of a bacteria population which is modeled by P = 50e^(0.2t)?

Correct Answer: 20%

Fill in the Blank Questions

Question 1:

The formula for exponential growth is y = A(1 + r)^t, where r represents the __________.

Correct Answer: growth rate

Question 2:

In the half-life formula, the variable 'H' represents the _________.

Correct Answer: half-life

Question 3:

To solve for an exponent in an exponential equation, we use _________.

Correct Answer: logarithms

Question 4:

A decay factor is a number between 0 and _________.

Correct Answer: 1

Question 5:

The initial amount in an exponential model is often referred to as the ________ amount.

Correct Answer: principal

Question 6:

The formula A = P(1 - r)^t represents exponential _________.

Correct Answer: decay

Question 7:

The property of logarithms that allows you to bring an exponent down is called the _________ property.

Correct Answer: power

Question 8:

If a substance doubles every 5 hours, its _________ time is 5 hours.

Correct Answer: doubling

Question 9:

The base of the natural logarithm is _________.

Correct Answer: e

Question 10:

The logarithm with base 10 is called the _________ logarithm.

Correct Answer: common

Educational Standards

A2.A.1.2:Use mathematical models to represent exponential relationships, such as compound interest, depreciation, and population growth. Solve these equations algebraically or graphically (including graphing calculator or other appropriate technology). A2.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms. A2.F.1.4:Graph exponential and logarithmic functions. Identify the domain, range, asymptotes, and x- and y-intercepts using various methods and tools that may include calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically.

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