SECTION 4.3SECTION 4.3

EXPONENTIAL FUNCTIONSEXPONENTIAL FUNCTIONS

EXPONENTIAL FUNCTIONS

EXPONENTIAL FUNCTIONS

An exponential function is a An exponential function is a function in the form:function in the form:

f(x) = bf(x) = b x x

where b > 0 and b where b > 0 and b 1 1

REAL-WORLD APPLICATIONSREAL-WORLD

APPLICATIONS

Population GrowthPopulation Growth

Radioactive DecayRadioactive Decay

GRAPH:GRAPH:

y = 2 y = 2 xx y = 3 y = 3 xx y y = 4 = 4 xx

x

41y

x

31y

x

21y

OBSERVATIONS ABOUT f(x) = b x

OBSERVATIONS ABOUT f(x) = b x

1.1. Domain:Domain: all realsall realsRange:Range: { y { y y > 0 } y > 0 }

2.2. f(x) is increasing for b > 1 f(x) is increasing for b > 1 f(x) is decreasing for f(x) is decreasing for

0 < b < 1.0 < b < 1.3.3. When f(x) is an increasing When f(x) is an increasing

function, the growth is function, the growth is more more rapid for larger rapid for larger values of b.values of b.

OBSERVATIONS, CONT.

OBSERVATIONS, CONT.

4.4. The graph of f(x) goesThe graph of f(x) goesthrough through the point (0,1) and the point (0,1) and y = 0 is a y = 0 is a horizontal horizontal asymptote.asymptote.

EXAMPLE:EXAMPLE:

Using transformations, Using transformations, describe how the graph of describe how the graph of g(x) = 1.5 g(x) = 1.5 x - 4x - 4 - 5 compares to - 5 compares to the graph of f(x) = 1.5the graph of f(x) = 1.5 x x

The graph of g(x) has the The graph of g(x) has the same shape as the graph of same shape as the graph of f(x), but is shifted 4 units to f(x), but is shifted 4 units to the right and 5 units down.the right and 5 units down.

EXAMPLE:EXAMPLE:

Using transformations, Using transformations, describe how the graph of describe how the graph of g(x) = 5g(x) = 5-x -x compares to the compares to the graph of f(x) = 5graph of f(x) = 5 x x

The graph of g(x) has the The graph of g(x) has the same shape as the graph of same shape as the graph of f(x), but is reflected across f(x), but is reflected across the y-axis.the y-axis.

x

51 x5

1 x5

THE NUMBER eTHE NUMBER e

The number The number ee is defined as is defined as the number that the the number that the expressionexpression n

n1

1

approaches as n approaches as n . In . In calculus, we express this using calculus, we express this using limit notation.limit notation. n

n1

1 lim e

n n

THE NUMBER eTHE NUMBER e

Look at Table 5Look at Table 5

THE NATURAL EXPONENTIAL

FUNCTION

THE NATURAL EXPONENTIAL

FUNCTION

f(x) = e f(x) = e xx

Find the value of e on your Find the value of e on your calculator. calculator.

e e 2.71828 2.71828

Graph:Graph:

f(x) = 2 f(x) = 2 xx g(x) = e g(x) = e xx h(x) = 3 h(x) = 3 xx

Which graph is that of g(x)?Which graph is that of g(x)? The one in the middle because e The one in the middle because e

is between 2 and 3.is between 2 and 3.

EXPONENTIAL EQUATIONS

EXPONENTIAL EQUATIONS

If aIf auu = a = avv, then u = v, then u = v

EXPONENTIAL EQUATIONS

EXPONENTIAL EQUATIONS

Solve:Solve: 33x + 1x + 1 = 81 = 81

Now do examples 7 & 8Now do examples 7 & 8

CONCLUSION OF SECTION 4.3CONCLUSION OF SECTION 4.3