intro to exponential functions lesson 3.1. contrast linear functions change at a constant rate rate...
TRANSCRIPT
![Page 1: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/1.jpg)
Intro to Exponential Functions
Lesson 3.1
![Page 2: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/2.jpg)
Contrast
LinearFunctions
• Change at a constant rate• Rate of change (slope) is a constant
ExponentialFunctions
• Change at a changing rate• Change at a constant percent rate
View differences using spreadsheet
View differences using spreadsheet
![Page 3: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/3.jpg)
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
![Page 4: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/4.jpg)
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
Year Option A Option B
1 $30,000 $40,000
2 $31,800 $41,200
3 $33,708 $42,400
4 $35,730 $43,600
5 $37,874 $44,800
6 $40,147 $46,000
7 $42,556 $47,200
8 $45,109 $48,400
9 $47,815 $49,600
10 $50,684 $50,800
11 $53,725 $52,000
12 $56,949 $53,200
13 $60,366 $54,400
14 $63,988 $55,600
![Page 5: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/5.jpg)
Example
• Consider a savings account with compounded yearly income You have $100 in the account You receive 5% annual interest
At end of year
Amount of interest earned
New balance in account
1 100 * 0.05 = $5.00 $105.00
2 105 * 0.05 = $5.25 $110.25
3 110.25 * 0.05 = $5.51 $115.76
4
5
View completed table
![Page 6: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/6.jpg)
Compounded Interest
• Completed table
At end of year
Amount of interest earned
New balance in account
0 0 $100.001 $5.00 $105.002 $5.25 $110.253 $5.51 $115.764 $5.79 $121.555 $6.08 $127.636 $6.38 $134.017 $6.70 $140.718 $7.04 $147.759 $7.39 $155.1310 $7.76 $162.89
![Page 7: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/7.jpg)
Compounded Interest
• Table of results from calculator Set y= screen
y1(x)=100*1.05^x Choose Table (Diamond Y)
• Graph of results
![Page 8: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/8.jpg)
Exponential Modeling
• Population growth often modeled by exponential function
• Half life of radioactive materials modeled by exponential function
![Page 9: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/9.jpg)
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
![Page 10: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/10.jpg)
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 hourAt end of hour Amount remaining
1 100 – 0.15 * 100 = 85
2 85 – 0.15 * 85 = 72.25
3
4
5
Fill in the rest of the
table
Fill in the rest of the
tableWhat is the
growth factor?
What is the growth factor?
![Page 11: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/11.jpg)
Decreasing Exponentials
• Completed chart
• Graph
At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06
At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06
Amount Remaining
0.00
20.00
40.00
60.00
80.00
100.00
0 1 2 3 4 5 6 7 8
At End of Hour
Mg
rem
ain
ing
Amount Remaining
0.00
20.00
40.00
60.00
80.00
100.00
0 1 2 3 4 5 6 7 8
At End of Hour
Mg
rem
ain
ing
Growth Factor = 0.85
Note: when growth factor < 1, exponential is a decreasing
function
Growth Factor = 0.85
Note: when growth factor < 1, exponential is a decreasing
function
![Page 12: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/12.jpg)
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
( ) 100 0.85 5.0tM t
![Page 13: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/13.jpg)
General Formula
• All exponential functions have the general format:
• Where A = initial value B = growth factor t = number of time periods
( ) tf t A B
![Page 14: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/14.jpg)
Typical Exponential Graphs
• When B > 1
• When B < 1
( ) tf t A B
View results of B>1, B<1 with spreadsheet
View results of B>1, B<1 with spreadsheet
![Page 15: Intro to Exponential Functions Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649e3a5503460f94b2bec9/html5/thumbnails/15.jpg)
Assignment
• Lesson 3.1A• Page 112• Exercises
1 – 23 odd
• Lesson 3.1B• Pg 113• Exercises
25 – 37 odd