Exponentials
Site:  TBAISD Moodle 
Course:  Michigan Algebra II Preview 2012 
Book:  Exponentials 
Printed by:  Guest user 
Date:  Sunday, July 5, 2020, 04:52 AM 
Growth
The exponential parent function is y = k^{x} where k is a constant. The easiest way to begin looking at this pattern is to make a table.
Recall from the Families of Functions Unit that the function g(x) = 2^{x} would create the following table and graph:
Notice that the graph increases more rapidly as x increases. The left side of the graph is approaching, but never touches the line y = 0, which is the graph's asymptote. Therefore, the domain of this function is and the range of the function is .
Decay
Recall that the function would create the following table and graph:
Notice that the graph decreases more rapidly as x increases. The right side of the graph is approaching, but never touches the line y = 0, which is the graph's asymptote. Therefore, the domain of this function is and the range of the function is .
Transformations
The transformations of functions that were presented in the Families of Functions unit also apply to exponential functions. The following list shows how the patterns apply to exponentials:
1. y = 2 ^{x} + a translates the graph a units up; asymptote equation y = a
2. y = 2 ^{x }a translates the graph a units down; asymptote equation y = a
3. y = 2 ^{xa }translates the graph a units right; asymptote equation y = 0
4. y = 2 ^{x+a}^{ }translates the graph a units left; asymptote equation y = 0
5. y=(2 ^{x}) reflects the graph across the xaxis; asymptote equation y = 0
6. y = 2 ^{x} reflects the graph across the yaxis; asymptote equation y = 0
Video Lessons
To learn how to transform exponential graphs, select the following link:
Translating Exponential Functions
To learn more about transforming exponential graphs, select the following link:
Stretching, Compressing & Reflecting Exponential Functions
Video Lesson:
To see more examples on exponential transformations, view the video below.
Identifying
Exponential Functions follow the form y = ab^{x} where a is the yintercept and b is the common ratio of the table. When modeling exponential data the avalue is the yintercept, therefore find the point where x = 0. To determine the bvalue calculate the common ratio between consecutive yvalues.
Modeling A Graph
In order to write the function that models an exponential graph, find at least two points on the graph. If the yintercept (0, a) is easily determined, select it as one of the two points. Otherwise, select two points with integer coordinates to make the computation easier. Then calculate the a and bvalues.
Example Write a function to model the graph.
Step 1. Determine the yintercept.
Notice (0, 4) is on the graph, therefore a = 4 and y = 4 ^{.} b^{x}
Step 2. Determine another point on the graph and substitute it into x and y in the equation, then solve for b.
Notice (1, 8) is on the graph, use it in the equation.
8 = 4 ^{.} b^{1}
2 = b
Therefore, y = 4 ^{. }2^{x }models this graph.
Modeling Points
When given two or more points of an exponential graph, try one of the following methods:
Example Find the exponential function that models the points (2, 6) and (4, 54)
Step 1. Put the points into a table.
x 
y 
0 

1 

2 
6 
3 

4 
54 
Step 2. Find the common multiplier between the yvalues.
6 ^{.} __ = 54
6 ^{.} 9 = 54
Example Continued
Step 3. Find the common ratio.
The common ratio is based on xvalues with an increment of 1. If the increment is not 1, then set up the equation
b ^{xincrement} = common multiplier and solve for b.
Guided Practice
To solidify your understanding of identifying and modeling exponential data, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Guided PracticeSources
Sources used in this book:Edwards, P. "The Exponential Function Applet." http://mathinsite.bmth.ac.uk/applet/exponential/exp.html (accessed 7/13/2010).
Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project
"Exponential Growth Equations and Graphs ." http://www.mathwarehouse.com/exponentialgrowth/graphand equation.php (accessed 7/13/2010).
Holt, Rinehart & Winston, "Exponential and Logarithmic Functions." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch 07_08_hom eworkhelp.html (accessed 7/13/2010).
Holt, Rinehart & Winston, "Identifying Exponential Data." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.ht ml?contentSr c=6468/6468.xml (accessed 7/13/2010).
Holt, Rinehart & Winston, "Transformations of Exponential Functions." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.ht ml?contentSr c=7157/7157.xml (accessed 7/13/2010).
Holt, Rinehart & Winston, "Translating Exponential Functions." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.ht ml?contentSr c=6467/6467.xml (accessed 7/13/2010).