What is an exponential function? An exponential function has the form of y = ab x, a ≠ 0, b is a...

What is an exponential function?• An exponential function has the form of y = abx, a ≠ 0,

b is a positive number other than 1

Example: y = 2x or y =(½) x

• There are 2 kinds of exponential function:1. Exponential growth: b > 1 Example: y = 4x

2. Exponential decay : 0 < b < 1 Example: y =(⅛) x

What is an exponential function?• An exponential function has the form of y = abx, a ≠ 0,

b is a positive number other than 1

Example: y = 2x or y =(½) x

• There are 2 kinds of exponential function:1. Exponential growth: b > 1 Example: y = 4x

2. Exponential decay : 0 < b < 1 Example: y =(⅛) x

Tuesday April 15, 2014

7.1 Exponential Growth Functions7.1 Exponential Growth Functions

Objective: To solve and graph polynomial and radical functionsObjective: To solve and graph polynomial and radical functionsEQ: Can you graph an exponential function?EQ: Can you graph an exponential function?

Exponential Growth Function:Domain: all real numbersRange: y > 0

To graph:Step 1: Plot 2 points (0, a) and (1, b)Step 2: Find couple points left of (0, a) and right to

(1, b)Step 3: Make a smooth curve, the left of the curve

never crosses x axis

Exponential Growth Function:Domain: all real numbersRange: y > 0



never crosses x axisExample 1: Graph y = 2x

Solution:a = 1, b = 2Step 1: Plot 2 points (0, 1) and (1, 2) Step 2: Points left of (0, 1) , right of (1, 2)

Step 3: Make a smooth curve

Example 1: Graph y = 2x

Solution:a = 1, b = 2Step 1: Plot 2 points (0, 1) and (1, 2) Step 2: Points left of (0, 1) , right of (1, 2)




Example:

Solution:

Example:

Solution:

Practice:

Solution:

Practice:

Solution:



Exponential Growth Models: y = a (1+r) t

a = initial amount, r = % (in decimal form), t = time

Example: In 1996, there were 2573 computer viruses security incidents. During the next 7 years, the number of incidents increased by 92% per year.1.Write the exponential growth model giving the number n of incidents t years after 1996 2.Graph the model3.Use the graph to estimate the year when there were about 125,000 computer security incidents

Solution:




Solution:

TranslationsTo graph a function of the form y = abx-h +k:• sketch y = abx

• then translate:h units horizontally and k units vertically

Example: y = 4. 2x-1 – 3

Graph y= 4. 2x

a = 4, b=2

h = 1, k = -3

TranslationsTo graph a function of the form y = abx-h +k:• sketch y = abx

• then translate:h units horizontally and k units vertically

Example: y = 4. 2x-1 – 3

Graph y= 4. 2x

a = 4, b=2

h = 1, k = -3



Example : Graph the function. State the domain and range1. y = 4x 2. y = ½ 3x 3. y

= 3x-1 – 3

Example : Graph the function. State the domain and range1. y = 4x 2. y = ½ 3x 3. y

= 3x-1 – 3




Solution:




Solution:

Wednesday April 23, 2014

7.2 Exponential Decay Functions7.2 Exponential Decay Functions


Exponential Decay Function has the form of f (x) = of y = abx, where 0 < b < 1

Domain: all real numbersRange: y > 0



never crosses x axis

Exponential Decay Function has the form of f (x) = of y = abx, where 0 < b < 1

Domain: all real numbersRange: y > 0



never crosses x axisExample 1: Graph y = (1/2)x

Solution:a = 1, b = 1/2Step 1: Plot 2 points (0, 1) and (1, 1/2) Step 2: Points left of (0, 1) , right of (1, 1/2)


Example 1: Graph y = (1/2)x

Solution:a = 1, b = 1/2Step 1: Plot 2 points (0, 1) and (1, 1/2) Step 2: Points left of (0, 1) , right of (1, 1/2)


Example:Example:

7.2 Exponential Decay Functions7.2 Exponential Decay Functions

TranslationsTranslations

Wednesday April 23, 2014



Exponential Decay Models: y = a (1 - r) t


Example: A snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.1.Write the exponent decay model?2.Find the value after 3 years3.Graph the model4.Find when the value of the snowmobile will be $2500

Solution:

Exponential Decay Models: y = a (1 - r) t


Example: A snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.1.Write the exponent decay model?2.Find the value after 3 years3.Graph the model4.Find when the value of the snowmobile will be $2500

Solution:

Thursday April 24, 2014

What is the Euler number or the Natural Base e?What is the Euler number or the Natural Base e?




Example: Example:

Calculating Compounded Interest?Calculating Compounded Interest?



Example: You have a $4,000,000 in an account that pay 6% annual interest compounded continuously. What is the balance after 1 year?

Solution:Compound annually A = P [1+ (.06)] 0.06 =Compound quarterly A = P [1+ (.06/4)] 06(4) =Compound monthly A = P [1+ (.06/12)] 06(12) =Compound daily A = P [1+ (.06/365)] 06(365) =Compound continuously: A = Pert = 4000e.06(1) =

Example: You have a $4,000,000 in an account that pay 6% annual interest compounded continuously. What is the balance after 1 year?

Solution:Compound annually A = P [1+ (.06)] 0.06 =Compound quarterly A = P [1+ (.06/4)] 06(4) =Compound monthly A = P [1+ (.06/12)] 06(12) =Compound daily A = P [1+ (.06/365)] 06(365) =Compound continuously: A = Pert = 4000e.06(1) =

Friday May 2, 2014

Rewrite the followings in exponential form(use whole numbers, both positive or negative)

8 = 0 = 12 =¼ =

DefinitionDefinition


7.4 Evaluate and Graph Logarithm Functions7.4 Evaluate and Graph Logarithm Functions

Objective: To solve and graph polynomial and radical functionsObjective: To solve and graph polynomial and radical functionsEQ: How do exponent and logarithm functions relate?EQ: How do exponent and logarithm functions relate?

Example: Example:

Example:

Example:

Inverse Functions:

f(x) = bx is the inverse function of g(x) = logbxTherefor:

Inverse Functions:

f(x) = bx is the inverse function of g(x) = logbxTherefor:



Example:

Example:

Example:

Example:

Parent Graphs for Logarithmic Functions:Parent Graphs for Logarithmic Functions:



Example:

Example:

Translations in Logarithmic Function Graph:

From the parent function: y = logb x to y = logb (x – h) + k

(the graph moves h units to the right and k units up)

Translations in Logarithmic Function Graph:

From the parent function: y = logb x to y = logb (x – h) + k

(the graph moves h units to the right and k units up)



Example: y = logb (x + 3) + 1

Solution:

Example: y = logb (x + 3) + 1

Solution:

What is an exponential function? An exponential function has the form of y = ab x, a ≠ 0, b is a...

Documents

Transcript of What is an exponential function? An exponential function has the form of y = ab x, a ≠ 0, b is a...