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 (think stretch/shrink).
Requirements:b greater than zero and not equal to onex is any real number
xbaxf )(
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).
)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(
…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)(
Graph the exponential function. xxf 3)(
x y
0 1
1 .3333
2 .1111
-1 3
-2 9
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Precalculus – Section 3.1
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
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
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
Graph the function.xexg 3)(
x f (x)
0 3
1 8.155
2 22.167
-1 1.103
-2 0.406
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!
Precalculus – Section 3.1
To find interest that is compounded continuously:
P = principle (amount invested)r = interest rate (as a decimal)t = time (in years)
rtPeA
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
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
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
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