Intro to exponential functions
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Intro to Exponential Functions. Lesson 3.1. Contrast. View differences using spreadsheet. Contrast. Suppose you have a choice of two different jobs at graduation Start at $30,000 with a 6% per year increase Start at $40,000 with $1200 per year raise Which should you choose?

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Intro to Exponential Functions

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Intro to exponential functions

Intro to Exponential Functions

Lesson 3.1


Contrast

Contrast

View differences using spreadsheet


Contrast1

Contrast

  • Suppose you have a choice of two different jobs at graduation

    • Start at $30,000 with a 6% per year increase

    • Start at $40,000 with $1200 per year raise

  • Which should you choose?

    • One is linear growth

    • One is exponential growth


Which job

Which Job?

  • How do we get each nextvalue for Option A?

  • When is Option A better?

  • When is Option B better?

  • Rate of increase a constant $1200

  • Rate of increase changing

    • Percent of increase is a constant

    • Ratio of successive years is 1.06


Example

Example

  • Consider a savings account with compounded yearly income

    • You have $100 in the account

    • You receive 5% annual interest

View completed table


Compounded interest

Compounded Interest

  • Completed table


Compounded interest1

Compounded Interest

  • Table of results from calculator

    • Set y= screen y1(x)=100*1.05^x

    • Choose Table (Diamond Y)

  • Graph of results


Exponential modeling

Exponential Modeling

  • Population growth often modeled by exponential function

  • Half life of radioactive materials modeled by exponential function


Growth factor

Growth Factor

  • Recall formulanew balance = old balance + 0.05 * old balance

  • Another way of writing the formulanew balance = 1.05 * old balance

  • Why equivalent?

  • Growth factor: 1 + interest rate as a fraction


Decreasing exponentials

Decreasing Exponentials

  • Consider a medication

    • Patient takes 100 mg

    • Once it is taken, body filters medication out over period of time

    • Suppose it removes 15% of what is present in the blood stream every hour

Fill in the rest of the table

What is the growth factor?


Decreasing exponentials1

Decreasing Exponentials

  • Completed chart

  • Graph

Growth Factor = 0.85

Note: when growth factor < 1, exponential is a decreasing function


Solving exponential equations graphically

Solving Exponential Equations Graphically

  • For our medication example when does the amount of medication amount to less than 5 mg

  • Graph the functionfor 0 < t < 25

  • Use the graph todetermine when


General formula

General Formula

  • All exponential functions have the general format:

  • Where

    • A = initial value

    • B = growth factor

    • t = number of time periods


Typical exponential graphs

Typical Exponential Graphs

  • When B > 1

  • When B < 1

View results of B>1, B<1 with spreadsheet


Assignment

Assignment

  • Lesson 3.1A

  • Page 112

  • Exercises1 – 23 odd

  • Lesson 3.1B

  • Pg 113

  • Exercises25 – 37 odd


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