exponential functions. objectives to use the properties of exponents to: simplify exponential...
TRANSCRIPT
Exponential FunctionsExponential Functions
ObjectivesObjectives
To use the properties of exponents to:To use the properties of exponents to: Simplify exponential expressions.Simplify exponential expressions. Solve exponential equations.Solve exponential equations.
To sketch graphs of exponential functions.To sketch graphs of exponential functions.
Exponential FunctionsExponential Functions
A polynomial function has the basic form: A polynomial function has the basic form: f f ((xx) = ) = xx33
An exponential function has the basic form: An exponential function has the basic form: f f ((xx) = 3) = 3xx
An exponential function has the variable in the An exponential function has the variable in the exponentexponent, , notnot in the in the basebase..
General Form of an Exponential Function: General Form of an Exponential Function: f f ((xx) = ) = NNxx, N > , N > 00
Properties of ExponentsProperties of Exponents
X YA A X YA
XYA
X YA
YXA
X
Y
A
A
XAB X XA B
XA
B
X
X
A
B
XA 1
XA
1XA XA
XYA Y XA X
Y A
Properties of ExponentsProperties of Exponents
2 32 2 52 32
2 62 2 424
1
2
1
16
232 62 64
Simplify:Simplify:
Properties of ExponentsProperties of Exponents3
2
3
3
3
2
3
3
3
3
2
7
9
3
3 23
2
1
3 1
9
1 12 22 8 1
216 16
27
8
4 1
22 8
Simplify:Simplify:
Exponential EquationsExponential Equations
Solve:Solve: Solve:Solve:5 125x 35 5x
3x
12( 1)7 7x
121x
12x
Exponential EquationsExponential Equations
8 4x
3 22 2x
3 2x
23x
3 22 2x
8 2x
3 12 2x
3 1x
13x
3 12 2x
Solve:Solve: Solve:Solve:
Exponential EquationsExponential Equations1
3 27x
13
3327x
19,683x
13 27
x
13 27x
3x
3x
33 3x
Not considered an exponential equation, because the variable
is now in the base.
Solve:Solve: Solve:Solve:
Exponential EquationsExponential Equations3
4 8x
4
3 4334 8x
43 8x
42x
16x
Not considered an exponential equation, because the variable
is in the base.
Solve:Solve:
Exponential FunctionsExponential Functions
General Form of an Exponential Function: General Form of an Exponential Function: f f ((xx) = ) = NNxx, N > , N > 00
gg((xx) = 2) = 2xx
x
22xx
gg(3) =(3) =
gg(2) =(2) =
gg(1) =(1) =
gg(0) =(0) =
gg(–1) =(–1) =
gg(–2) =(–2) =
8
4
2
112 1
222
212
14
Exponential FunctionsExponential Functions
gg((xx) = 2) = 2xx
x
2
1
0
4
2
1
g x
–1
–2
12
14
Exponential FunctionsExponential Functions
gg((xx) = 2) = 2xx
Exponential FunctionsExponential Functions
hh((xx) = 3) = 3xx
x
2
1
0
9
3
1
h x
–1
–2
13
19
Exponential FunctionsExponential Functions
hh((xx) = 3) = 3xx
Exponential FunctionsExponential Functions
Exponential functions with Exponential functions with positive bases positive bases greatergreater than than 1 have graphs that are 1 have graphs that are increasingincreasing..
The function never crosses The function never crosses the the xx-axis because there is -axis because there is nothing we can plug in for nothing we can plug in for xx that will yield a zero that will yield a zero answer.answer.
The The xx-axis is a left -axis is a left horizontal asymptote.horizontal asymptote.
hh((xx) = 3) = 3x x (red)(red)
gg((xx) = 2) = 2xx (blue) (blue)
Exponential FunctionsExponential Functions
A A smallersmaller base means base means the graph rises more the graph rises more graduallygradually..
A A largerlarger base means the base means the graph rises more graph rises more quicklyquickly..
Exponential functions will Exponential functions will notnot have have negativenegative bases. bases.hh((xx) = 3) = 3x x (red)(red)
gg((xx) = 2) = 2xx (blue) (blue)
The Number The Number ee
e
2.71828169 A base often associated with exponential functions is:
The Number The Number ee
Compute:Compute: 1
0lim 1 x
xx
x
–.1
–.01
–.001
2.868
2.732
2.7196
1
1 xx
1
0lim 1 x
xx
x
.1
.01
.001
2.5937
2.7048
2.7169
1
1 xx
1
0lim 1 x
xx
2.71828169
The Number The Number ee
Euler’s number Euler’s number
Leonhard Euler Leonhard Euler (pronounced “oiler”)(pronounced “oiler”)
Swiss mathematician Swiss mathematician and physicistand physicist
The Exponential FunctionThe Exponential Function
f f ((xx) = ) = eexx
Exponential FunctionsExponential Functions
x
2
1
0
4
2
1
j x
–1
–2
12
14
12
xj x
Exponential FunctionsExponential Functions 1
2
xj x
Exponential functions with positive bases less than 1 have graphs that are decreasing.
Why study exponential functions?Why study exponential functions?
Exponential functions are used in our real world Exponential functions are used in our real world to measure growth, interest, and decay.to measure growth, interest, and decay.
Growth obeys exponential functions.Growth obeys exponential functions.Ex: rumors, human population, bacteria, Ex: rumors, human population, bacteria, computer technology, nuclear chain reactions, computer technology, nuclear chain reactions, compound interestcompound interest
Decay obeys exponential functions.Decay obeys exponential functions.Ex: Carbon-14 dating, half-life, Newton’s Law of Ex: Carbon-14 dating, half-life, Newton’s Law of CoolingCooling