Lesson 3.1, page 376 Exponential Functions
description
Transcript of Lesson 3.1, page 376 Exponential Functions
![Page 1: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/1.jpg)
Lesson 3.1, page 376 Exponential Functions
Objective: To graph exponentials equations and functions, and solve
applied problems involving exponential functions and their
graphs.
![Page 2: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/2.jpg)
Look at the following…
Polynomial Exponential
2( ) 4 3 1 ( ) 4 3xf x x x f x
![Page 3: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/3.jpg)
Real World Connection Exponential functions are used to
model numerous real-world applications such as population growth and decay, compound interest, economics (exponential growth and decay) and more.
![Page 4: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/4.jpg)
REVIEW Remember: x0 = 1 Translation – slides a figure
without changing size or shape
![Page 5: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/5.jpg)
Exponential Function
The function f(x) = bx, where x is a real number, b > 0 and b 1, is called the exponential function, base b.
(The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers.)
![Page 6: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/6.jpg)
Examples ofExponential Functions, pg. 376
1( ) 3 ( )3
( ) (4.23)
xx
x
f x f x
f x
![Page 7: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/7.jpg)
See Example 1, page 377. Check Point 1: Use the function
f(x) = 13.49 (0.967) x – 1to find the number of О-rings expected to fail
at a temperature of 60° F. Round to the nearest whole number.
![Page 8: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/8.jpg)
Graphing Exponential Functions
1. Compute function values and list the results in a table.
2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.
![Page 9: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/9.jpg)
Check Point 2 -- Graph the exponential function y = f(x) = 3x.
(3,1/27)1/273
(2, 1/9)1/92
(1, 1/3)1/31
(3, 27)273
9
3
1
y = f(x) = 3x
(2, 9)2
(1, 3)1
(0, 1)0
(x, y)x
![Page 10: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/10.jpg)
Check Point 3: Graph the exponential function
(3,1/27)1/273(2, 1/9)1/92(1, 1/3)1/31(3, 27)273
931
(2, 9)2(1, 3)1(0, 1)0(x, y)x 1( )
3
x
y f x
1( )3
x
y f x
![Page 11: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/11.jpg)
Characteristics of Exponential Functions, f(x) = bx, pg. 379
Domain = (-∞,∞) Range = (0, ∞) Passes through the point (0,1) If b>1, then graph goes up to the right and
is increasing. If 0<b<1, then graph goes down to the right
and is decreasing. Graph is one-to-one and has an inverse. Graph approaches but does not touch x-axis.
![Page 12: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/12.jpg)
Observing Relationships
![Page 13: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/13.jpg)
Connecting the Concepts
![Page 14: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/14.jpg)
Example -- Graph y = 3x +
2.The graph is that of y = 3x shifted left 2 units.
24338122719031
1/3y= 3 x+2
123x
![Page 15: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/15.jpg)
Example: Graph y = 4 3x
3.9633.8823.671
301523y
123x
The graph is a reflection of the graph of y = 3x across the y-axis, followed by a reflection across the x-axis and then a shift up of 4 units.
![Page 16: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/16.jpg)
The number e (page 381) The number e is an irrational
number. Value of e 2.71828 Note: Base e exponential functions
are useful for graphing continuous growth or decay.
Graphing calculator has a key for ex.
![Page 17: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/17.jpg)
Practice with the Number e Find each value of ex, to four decimal
places, using the ex key on a calculator.a) e4 b) e0.25
c) e2 d) e1
Answers:a) 54.5982 b) 0.7788c) 7.3891 d) 0.3679
![Page 18: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/18.jpg)
Natural Exponential Function
Remember e is a number
e lies between 2 and 3
xf (x) e
![Page 19: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/19.jpg)
Compound Interest Formula
A = amount in account after t years P = principal amount of money
invested R = interest rate (decimal form) N = number of times per year interest
is compounded T = time in years
1 ntrA P
n
![Page 20: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/20.jpg)
Compound Interest Formula for Continuous Compounding
A = amount in account after t years P = principal amount of money
invested R = interest rate (decimal form) T = time in years
rtA Pe
![Page 21: Lesson 3.1, page 376 Exponential Functions](https://reader036.fdocuments.net/reader036/viewer/2022062408/56815be4550346895dc9d707/html5/thumbnails/21.jpg)
See Example 7, page 384.Compound Interest Example
Check Point 7: A sum of $10,000 is invested at an annual rate of 8%. Find the balance in that account after 5 years subject to a) quarterly compounding and b) continuous compounding.