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Lecture Notes on Exponential Function | MATH 1130, Study notes of Algebra

Material Type: Notes; Class: College Algebra; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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Name:
Date:
Instructor:
Notes for 4.2 Exponential Functions (pp. 402 – 414)
I. Exponents and Their Properties (pp. 402 – 404)
Def. An exponential function has its variable as an exponent.
It is written as ( ) x
f
xa=, where a > 0, a
1 and x is any real number.
Ex. Recall: 38x= vs. 2 8
x
=
is an exponential equation.
*Calculator reminders:
To put an exponent into the calculator, use the ^ key. If the exponent is a fraction, you
must put parentheses around the entire fraction so that the calculator will “see” the entire
exponent.
Ex. 3.14
2 = 2 ^ 3.14 ENTER and
3
2
2= 2 ^ (3 / 2) ENTER
Ex. If ( ) 2x
fx=, find f(-1), f(3), f ( 5/2), f(4.92) Use calculator.
Recall Properties of Exponents:
1. mn mn
aa a
+
=i
2.
()
n
mmn
aa=
3. 01a=
4. 1
m
m
aa
=
5. 1 1
anything =
II. Exponential Functions (pp. 404 – 407)
A. For ( ) x
f
xa=,where a > 1. (p. 405) Ex. ( ) 2x
fx
=
. Graph is an upwards swoop.
Characteristics of the exponential when a (the base) > 1:
1. Domain:
()
,−∞ and Range:
(
)
0,
graph lies on top of the x-axis.
2. Graph has no x intercept because it doesn’t cross the x axis… only approaches it.
3. Graph has a y intercept at (0, 1), since 0
21
=
(in fact, 01a
, so that point is on all
exponential functions!!)
4. Graph is increasing.
5. Graph is continuous.
6. Any graph contains the points (0, 1), (1,a), and (-1, 1
a).
7. The larger the value of the base, the steeper the graph.
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Name: Date: Instructor:

Notes for 4.2 Exponential Functions (pp. 402 – 414)

I. Exponents and Their Properties (pp. 402 – 404)

Def. An exponential function has its variable as an exponent.

It is written as f ( ) x = ax , where a > 0, a ≠ 1 and x is any real number.

Ex. Recall: x^3^ = 8 vs. 2 x^ = 8 is an exponential equation.

*Calculator reminders: To put an exponent into the calculator, use the ^ key. If the exponent is a fraction, you must put parentheses around the entire fraction so that the calculator will “see” the entire exponent.

Ex. 2 3.14 = 2 ^ 3.14 ENTER and

3 22 = 2 ^ (3 / 2) ENTER

Ex. If f ( ) x = 2 x , find f(-1), f(3), f ( 5/2), f(4.92) Use calculator.

Recall Properties of Exponents:

  1. am i an = a m^ + n

m n mn a = a

  1. a^0^ = 1

m^1 a (^) am

  1. 1 anything^ = 1

II. Exponential Functions (pp. 404 – 407)

A. For f ( ) x = ax ,where a > 1. (p. 405) Ex. f ( ) x = 2 x. Graph is an upwards swoop.

Characteristics of the exponential when a (the base) > 1:

1. Domain: ( −∞ ∞, ) and Range: ( 0, ∞ ) graph lies on top of the x-axis.

  1. Graph has no x intercept because it doesn’t cross the x axis… only approaches it.
  2. Graph has a y intercept at (0, 1), since 2 0 = 1 (in fact, a^0^ = 1 , so that point is on all exponential functions!!)
  3. Graph is increasing.
  4. Graph is continuous.
  5. Any graph contains the points (0, 1), (1,a), and (-1,

a

  1. The larger the value of the base, the steeper the graph.

B. For f ( ) x = ax , where a < 1 (fraction base) Ex.

x f x = ;

x g x = Graph is a

downwards swoop. Characteristics of an exponential, where the base ( a ) is < 1:

1. Domain: ( −∞ ∞, ) and Range: ( 0, ∞ ) graph lies on top of the x-axis.

  1. Graph has no x intercept because it doesn’t cross the x axis… only approaches it.
  2. Graph has a y intercept at (0, 1).
  3. Graph is decreasing.
  4. Graph is continuous.
  5. Any graph contains (0, 1), (1, a) and (-1,

a

  1. The smaller the value of a , the steeper the graph.

Translations still work. p. 407, Ex. 3 x 2 x^ − 2 x 2 x +^3 2 x^ + 3

  • 0 1 2 3

III. Solving Exponential Equations (pp. 407 - 408)

  1. This type of exponential uses the property that if a m^ = an (the bases are the same), then m = n (the exponents will be the same.

Ex. 3 x +^1 = 81

Since 81 = 3^4 , we can write the equation as 3 x +^1 = 34 Now that the bases (3) are the same, then x + 1 = 4 And x = 3

Ex. 2 7

x − (^) =

5 2 x −^7 = 5 −^1 (recall that a negative exponent will take care of a fraction) 2 x – 7 = − 1 2 x = 6 x = 3 Omit p. 408, Example 6.

Continuously compounded interest runs the compounding times faster and faster and faster, until it get to be “instantaneous”. The formula is: A = Pert , where A = Future Value, P = Present Value or Principal, r = rate as a decimal, and t = time in years. The e is that number 2.71828 that the ( ) always settles to…(demo in video).

Ex. For an investment of $1000 at 4% for 10 years, compare the different amounts earned when it is compounded in the chart below:

Compoundings $1 $ Annually Semiannually Quarterly Monthly Daily Continuously (use different Formula)

Assignments: Text: pp. 414 – 417#1 – 21 every other odd, 33, 38, 39 – 55 odd, 71, 73, 75ab, 77 - 80