[email protected] MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 1 Bruce Mayer, PE Chabot College...

[email protected] • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.1a§9.1aExponential FcnsExponential Fcns



Review §Review §

Any QUESTIONS About• §8.5 → Rational InEqualities

Any QUESTIONS About HomeWork• §8.5 → HW-41

8.5 MTH 55



Exponential FunctionExponential Function

A function, f(x), of the form

is called an EXPONENTIAL function with BASE a.

The domain of the exponential function is (−∞, ∞); i.e., ALL Real Numbers



Recall Rules of ExponentsRecall Rules of Exponents

Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

ax ay axy ,

ax

ay ax y ,

ab x axbx ,

ax yaxy ,

a0 1,

a x 1

ax 1

a

x

.



Evaluate Exponential FunctionsEvaluate Exponential Functions

Example

Solution

Example

Solution



Evaluate Exponential FunctionsEvaluate Exponential Functions

Example

Solution



Example Example Graph Graph yy = = ff((xx) =3) =3xx

Graph the exponential fcn: ( ) 3 .xf x

Make T-Table,& Connect Dots

x y0

1

–1

2

–2

3

1

3

1/3

9

1/9

27

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

6

2

5

1

-1

-2

78



Example Example Graph Exponential Graph Exponential

Graph the exponential fcn:


x y

0

1

–1

2

–2

–3

1

1/3

3

1/9

9

27• This fcn is a

REFLECTION of y = 3x

3xy

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

6

2

5

1

-1

-2

78

1( )

3

xy f x





Construct SideWays T-Table

x −3 −2 −1 0 1 2 3

y = (1/2)x 8 4 2 1 1/2 1/4 1/8

Plot Points and Connect Dots with Smooth Curve




As x increases in the positive direction, y decreases towards 0



Exponential Fcn PropertiesExponential Fcn Properties

Let f(x) = ax, a > 0, a ≠ 1. Then

A. The domain of f(x) = ax is (−∞, ∞).

B. The range of f(x) = ax is (0, ∞); thus, the entire graph lies above the x-axis.

C. For a > 1 (e.g., 7)i. f is an INcreasing function; thus, the graph

is RISING as we move from left to right

ii. As x→∞, y = ax increases indefinitely and VERY rapidly




Let f(x) = ax, a > 1, a ≠ 1. Then iii. As x→−∞, the values of y = ax get

closer and closer to 0.

D. For 0 < a < 1 (e.g., 1/5)i. f is a DEcreasing function; thus, the graph

is falling as we scan from left to right.

ii. As x→−∞, y = ax increases indefinitely and VERY rapidly

iii. As x→ ∞, the values of y = ax get closer and closer to 0




Let f(x) = ax, a > 0, a ≠ 1. Then

E. Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus:

i. 2121 xxaa xx

ii. f has an inverse




Let f(x) = ax, a > 0, a ≠ 1. Then

F. The graph f(x) = ax has no x-intercepts • In other words, the graph of f(x) = ax

never crosses the x-axis. Put another way, there is no value of x that will cause f(x) = ax to equal 0

G. The x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax.



Translate Exponential GraphsTranslate Exponential Graphs

Translation Equation Effect on Equation

HorizontalShift

y = ax+b

= f (x + b)Shift the graph of y = ax, b units(i) Left if b > 0.(ii) Right if b < 0.

VerticalShift

y = ax + b = f (x) + b

Shift the graph of y = ax, b units(i) Up if b > 0.(ii) Down if b < 0.



Example Example Sketch Graph Sketch Graph

By TranslationMove DOWNy = 3x by 3 Units

Note• Domain: (−∞, ∞)

• Range: (−4, ∞)

• Horizontal Asymptote: y = −4



Example Example Sketch Graph Sketch Graph

By TranslationMove LEFTy = 3x by 1 Unit

Note• Domain: (−∞, ∞)

• Range: (0, ∞)

• Horizontal Asymptote: y = 0



Alternative Graph: Swap Alternative Graph: Swap xx & & yy

It will be helpful in later work to be able to graph an equation in which the x and y in y = ax are interchanged

-6 -4 -2 0 2 4 6 8 10 12

-6

-4

-2

0

2

4

6

8

10

12

yx 3.2

xy 3.2



Example Example Graph Graph xx = 3 = 3yy



x y

1

3

1/3

9

1/9

27

0

1

–1

2

–2

3

x

y

-3 -2 -1 1 2 3 4 5 6 7 8 9

4

3

6

2

5

1

-1-2



Example Example Apply Exponential Apply Exponential

Example Bank Interest compounded annually.

The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula



Example Example Compound Interest Compound Interest

Suppose that $60,000 is invested at 5% interest, compounded annually

a) Find a function for the amount in the account after t years

SOLUTION

a) = $60000(1 + 0.05 )t

= $60000(1.05)t



Example Example Compound Interest Compound Interest

Suppose that $60,000 is invested at 5% interest, compounded annually

b) Find the amount of money amount in the account at t = 6.

SOLUTION

b) A(6) = $60000(1.05)6 $80,405.74



Example Example Bacterial Growth Bacterial Growth

A technician to the Great French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubled every hour.

Assume that the bacteria count B(t) is modeled by the equation

• Where t is time in hours




Given Bacterial Growth Equation

Find:a) the initial number of bacteria,

b) the number of bacteria after 10 hours; and

c) the time when the number of bacteria will be 32,000.




a) INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn:

B0 B 0 200020 20001 2000

b) At Ten Hours Sub t = 10 into Eqn:




c) Find t when B(t) = 32,000

Thus 4 hours after the starting time, the number of bacteria will be 32k

32000 20002t

16 2t

24 2t

4 t



WhiteBoard WorkWhiteBoard Work

Problems From §9.1 Exercise Set• 36, 40, 54

USAPersonalSavingsRate



All Done for TodayAll Done for Today

BacteriaGrowFAST!

• Note: 37 °C = 98.6 °F (Body Temperature)



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–



Irrational ExponentsIrrational Exponents

By The Properties of Exponents we Can Evaluate Bases Raised to Rational-Number Powers Such as

23

2

3

13 23

123

2

77777

What about expressions with IRrational exponents such as:

To attach meaning to this expression consider a rational approximation, r, for the Square Root of 2




Approximate byITERATION on:

1.4 < r < 1.5

1.41 < r < 1.42

1.414 < r < 1.415

1.4 1.515.245 7 7 18.520p 1.41 1.4215.545 7 7 18.850p

1.414 1.41515.666 7 7 15.697p

closes in on 2r 2 closes in on 7p

277

2

rp

r




Thus by Iteration

27 15.6728909 Any positive irrational exponent can be

interpreted in a similar way. Negative irrational exponents are then

defined using reciprocals.

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