MATH 1314 College Algebra Christy Dittmar

Section 5.3 notes: Exponential Functions and Models

Exponential Functions and Models (5.3)

Exponential Functions

General form f(x)= Ca*

1. Find C and a.

-1

10 120 110 so
10 = ca0 f(x) 10aY =

a= ↓yto select when x= 1

1:10 = C
20 10.a
=

+20 10a va= 2

= =

b.

-2
f(x) 10.2X
=

45 15 5/3 5/9
do not
f(x) = (aY f(x) 3.aY
multiply the
=

Istpt: X 0
=

2nd point: x= ( 1052.bC Of

5 C.a0 the X

tb aabj
=
=

=
a
C= S
C. f(x) 3.5*
=

never write
-
33333...
instead write
t

x 0f(x) 2
=
=

x = 1f(x) 1
=

f(x) CaY =
1= 2.d
I = a

2 c.
=

c= 2
f(x) 2.E
=

Page of 6
 College Algebra
Christy Dittmar
MATH 1314
Section 5.3 notes: Exponential Functions and Models

Applications

Exponential growth

Compound Interest

2. Example: $1000 invested at 5% compounded annually.

year

Investment $1000 1090 (102.3 1137.63

3
I
= PrT
↑
1000 1=1000..09.1
I =
1000.11.05)'
I = 1858
v = .03 1= 50
T 1 =

investment: 1058
3. Investment in year f:

I = 1000 (1.03)

Note: powers on a calculator A

Compound interest formula

Interest compounded n times per year: A =

P = Principal
A = amount after t years

7 = annual interest rate (convert to decimal)

n = number of compounding periods per year
number of years

4. A principal of $1500 is invested at an interest rate of 6.5%, compounded quarterly.

How much is present after five years? 4.S
P $1580
= A= 1500(1+.06
v 6.9% =.063%
$2070.63
=

A =

n4 =

tg =

Page 2 of 6
MATH 1314 College Algebra Christy Dittmar
notes: Exponential Functions and
Section 5.3 Models

The base e

10 2.59374246
100 2.70481383
1000 2.71692393
10000 2.71814593
100000 2.71826824

Powers of e on a calculator;
e

5. Approximate to three decimal places: 21.8

6.8496
Interest compounded continuously

Interest compounded continuously: A = Pet

6. How much money do you need to invest now, at an interest rate of 10.25%
compounded continuously, to have $100,000 in your account after 18 years? Round
to the nearest cent.

100000 = es(18)
↑

1.845 *
e e

P $13,002.63
=

Page 3 of 6
MATH 1314 College Algebra Christy Dittmar
Section 5.3 notes: Exponential Functions and Models
Graphing

7. f(x)=2*

·125.25 0.514
2 8

Graph:
To

O
a f

T
⑧
CO

Characteristics of f(x)=a,a>1:
y-intercept = (0,1)

increasing fast in first quadrant (doubling time)

x-axis is horizontal asymptote

Page 4 of 6
MATH 1314 College Algebra Christy Dittmar
Section 5.3 notes: Exponential Functions and Models

8.

Graph:

j
at
*
-

Characteristics of g(x) = a* 0<a<1:

y-intercept= (0,1)

decreasing in first quadrant (half life)

x-axis is horizontal asymptote

Page 5 of 6
MATH 1314 College Algebra Christy Dittmar
Section 5.3 notes: Exponential Functions and Models

Exponential decay and half-life

Modeling with half-life A(x): A0 A is the initial amount; k is the

"half-life" (the time it takes for half the substance to remain)

Radon-222

9. Radon is a naturally occurring radioisotope. Radon-222 is an isotope of Radon, with a

half-life of 4 days.

a. What percent of the initial amount of Radon-222 released remains after one
week?

A = 10
grams
A =

10(t)"Y
A =
2.973
grams

b. If five milligrams of Radon-222 are released, how much remains after 28 days?

A =
3(t)-
A = .03906
milligrams

Page 6 of 6