Transcript What is an exponential function?

Tuesday April 15, 2014
7.1 Exponential Growth Functions
Objective: To solve and graph polynomial and radical functions
EQ: Can you graph an exponential function?
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
Exponential Growth Function:
Domain: all real numbers
Range: 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
Example 1: Graph y = 2x
Solution:
a = 1, b = 2
Step 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
Tuesday April 15, 2014
7.1 Exponential Growth Functions
Example:
Solution:
Practice:
Solution:
Tuesday April 15, 2014
7.1 Exponential Growth Functions
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 model
3. Use the graph to estimate the year when there were about
125,000 computer security incidents
Solution:
Tuesday April 15, 2014
7.1 Exponential Growth Functions
Translations
To 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 range
1. y = 4x
2. y = ½ 3x
3. y = 3x-1 – 3
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 model
3. Use the graph to estimate the year when there were about
125,000 computer security incidents
Solution:
Wednesday April 23, 2014
7.2 Exponential Decay Functions
Objective: To solve and graph polynomial and radical functions
EQ: Can you graph an exponential function?
Exponential Decay Function has the form of f (x) = of y = abx,
where 0 < b < 1
Domain: all real numbers
Range: 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
Example 1: Graph y = (1/2)x
Solution:
a = 1, b = 1/2
Step 1: Plot 2 points (0, 1) and (1, 1/2)
Step 2: Points left of (0, 1) , right of (1, 1/2)
Step 3: Make a smooth curve
Wednesday April 23, 2014
7.2 Exponential Decay Functions
Example:
Translations
Tuesday April 15, 2014
7.2 Exponential Growth Functions
Exponential Decay Models:
y = a (1 - r) t
a = initial amount, r = % (in decimal form), t = time
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 years
3. Graph the model
4. Find when the value of the snowmobile will be $2500
Solution:
Thursday April 24, 2014
Thursday April 24, 2014
7.3 Exponential Growth Functions
Objective: To solve and graph polynomial and radical functions
EQ: Can you graph an exponential function?
What is the Euler number or the Natural Base e?
Example:
Thursday April 24, 2014
7.3 Exponential Growth Functions
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
=
06(4)
Compound quarterly A = P [1+ (.06/4)]
=
06(12)
Compound monthly A = P [1+ (.06/12)]
=
06(365)
Compound daily
A = P [1+ (.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
=
=
=
=
Thursday April 24, 2014
7.4 Evaluate and Graph Logarithm Functions
Objective: To solve and graph polynomial and radical functions
EQ: How do exponent and logarithm functions relate?
Definition
Example:
Example:
Thursday April 24, 2014
7.4 Evaluate and Graph Logarithm Functions
Inverse Functions:
f(x) = bx is the inverse function of g(x) = logbx
Therefor:
Example:
Example:
Thursday April 24, 2014
7.4 Evaluate and Graph Logarithm Functions
Parent Graphs for Logarithmic Functions:
Example:
Thursday April 24, 2014
7.4 Evaluate and Graph Logarithm Functions
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: