![Page 1: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/1.jpg)
Precalculus – Section 3.1
![Page 2: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/2.jpg)
An exponential function is a function of the form
We call b the base of the exponential function.a is a constant multiplier (think stretch/shrink).
Requirements:b greater than zero and not equal to onex is any real number
xbaxf )(
![Page 3: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/3.jpg)
xxf 2)( xxf 21)(
)1,0(
The graph of the function crosses the y-axis at the point (0,a).
)2,1(
The graph also contains the point (1,a⋅b).
![Page 4: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/4.jpg)
)2,1(
)1,0(
To get more points on the graph:
As you increase the value of x by 1, you multiply the previous y-value by the base b.
As you decrease the value of x by 1, you divide the y-value by b.
)4,2(
)8,3(
)2
1,1(
![Page 5: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/5.jpg)
…then the function grows.
…then the function decays.
If the base is greater than 1…
If the base is less than 1…
x
xf
2
3)(
x
xf
3
1)(
![Page 6: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/6.jpg)
Graph the exponential function. xxf 3)(
x y
0 1
1 .3333
2 .1111
-1 3
-2 9
![Page 7: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/7.jpg)
p. 206: 1-4, 7-10, 19-24
![Page 8: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/8.jpg)
Precalculus – Section 3.1
![Page 9: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/9.jpg)
A common choice for the base of an exponential function is e .
e is the called the natural base because it naturally occurs in things such as:
compound interest radioactive decay science applications
![Page 10: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/10.jpg)
The value of e to 15 decimal places…
e = 2.718281828459045…
Think of President Andrew Jackson (the guy on the $20 bill)!
Good enough for precalculus use: 2.718281828
![Page 11: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/11.jpg)
Evaluate each function. Round to 3 decimal places.
1.
2.
1.4 when )( xexf x
-1 when )( 3 xexf x
1.4)1.4( ef 1.4)718281828.2(
340.60
231)1( eef
389.7
![Page 12: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/12.jpg)
Graph the function.xexg 3)(
x f (x)
0 3
1 8.155
2 22.167
-1 1.103
-2 0.406
![Page 13: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/13.jpg)
p. 206: 5,6, 25-30
You may want to scale your graphs to fit them on the paper.
Tomorrow: using exponential functions to problem solve!
![Page 14: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/14.jpg)
Precalculus – Section 3.1
![Page 15: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/15.jpg)
To find interest that is compounded continuously:
P = principle (amount invested)r = interest rate (as a decimal)t = time (in years)
rtPeA
![Page 16: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/16.jpg)
Find the current balance of a $7000 savings fund after 1 year if the interest is compounded continuously at 8%.
rtPeA)1)(08.0()7000( eA
01.7583$A
![Page 17: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/17.jpg)
To find the amount of a radioactive substance that remains after t years:
N = initial quantityt = time (in years)H = half-life of substance (in years)
H
t
NQ
2
1
![Page 18: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/18.jpg)
A certain radioactive substance has a half-life of 825 years. Find the amount of substance that remains after 1000 years if the initial amount is 50 pounds.
H
t
NQ
2
1
825
1000
2
150
Q
pounds 58.21Q
![Page 19: Precalculus – Section 3.1. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier](https://reader030.vdocuments.us/reader030/viewer/2022032722/56649f425503460f94c62b1e/html5/thumbnails/19.jpg)
p. 207: 51-53, 55-58