I am very excited about using this blog site to help my students. As we begin this new adventure, I am aware that many of you are experienced "bloggers". You may already participate in social networking sites sucy as Live Journal, My Space, Facebook, etc. These sites are basically journaling tools. Our Trig Blog will be very different - we will use blogging as an educational tool to share and build understanding of the concepts we discussi in class.
Last semester, I began this TRIG BLOG to allow an assigned student, the "Scribe", to post a review (summary) of the day's lesson. The Scribe will receive a quiz grade for his work. After reading the summary, other students may then post additions, corrections, or questions regarding the lesson or homework problems. Students who post additional information or answers to questions will receive an "early bird" credit.
"GROUND RULES" This site is strictly "business" for our spring trig class. It is not a forum for any personal issues. I expect all students to be professional when using this site. Any non professional uses will be disciplined according to our student handbook and code of conduct policies.
This site will be a great resource for all of you to get the extra help you need. Occasionally, I will "check in" at night and offer my help.
I will check the site each morning before school and note the questions and problems you are having with previous lessons. This will help me to know exactly what you need help with that day in class.
I'm looking forward to a great semester :)
Mrs. S
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You will write your summaries of Lesson 8.1a by leaving a "comment"
This is our first day on this blog
Hello Mrs. Snipes! It's a beautiful day in the neighborhood!
"The Original Summary of Chapter 8.1a" as told by trey-380.
In class we discussed and learned how to use exponential functions and logarithmic equations, which are used in everyday life to solve real life problems.
We also learned how to write exponential equations and sketch graphs of exponential functions.
An exponential funciton is the equation " y=ab^x ". Please also note that a CAN NOT equal 0, b >0, and b CAN NOT equal 1. If b>1 then your graph will be a growth growing from left and reaching higher as it goes right. If b<1 then your graph will be a decay falling left to right getting lower as it moves.
The variable a in the equation will always be your y-intercept on the graph and the b variable is known as the "stretch factor" due to the fact it can stretch towards the y-axis or away from the y-axis. The higher the b value, the closer to the y-axis your curve will be. In contrast, the lower the b value, the farther to the y-axis your curve will be.
Example:
(4,8) and (6,32)
First you will solve both equations for a.
8=ab^4 32=ab^6
8/b^4=a 32/b^6=a
Next set both equations equal to each other and solve for b.
8/b^4 = 32/b^6
8b^6 = 32b^4 (after you cross multiply to get rid of fractions)
8b^6/b^4 = 32b^4/b^4
8b^2/8 = 32/8
b^2 = 4
√b^2 = 2
Plug b back in to one of your equations and solve for a.
8/2^4 = a
8/16 = a
1/2 = a
You then plug a and b in to the entire equation and check!!
y= .5(2)^x
Lesson 8-1
With this lesson you will learn to use exponential functions and logarithmic equations to model and solve real life problems.
An Eponential Function is an equation of the form y=ab^x with the restrictions a is not equal to 0, b>0, and b is not equal to 1.
Some shortcuts for graphing y=ab^x
- a is always going to be the y-int. of the graph.
- b is known as the "stretch factor."
*hint* If 1 < b < 2 it will stretch furthur to the y-axis.
How to write an exponential function given two ordered pairs.
EX. (4,8), (6,32)
1. Take your ordered pairs and set them in the form of y=ab^x
2. Solve for a.
8/b^4=ab^4/b^4
a= 8/b^4
32/b^6=ab^6/b^6
a=32/b^6
3.Set a's equal to each other and solve for b.
8/b^4=32/b^6
= 8b^6/b^4=32b^4/b^4
=32/8=8b^2/8
b^2=4 *Take square root*
b=2.
4. Substitue b in for one of the original equations for a and find a.
a=8/2^4 a=.5
5. Put answer in y=ab^x form.
y=.5(2)^x
Some other terms that are mentioned in this lesson are
Asymptote- A line that a graph approached as x or y increases in absolute value.
Decay Factor- An exponential function in which b is less than 1 but greater than 0.
Lesson 8.1a
Explororing Exponential Models
-Learn how to write and graph exponential function
-Be able to explain how a function can show growth and decay
-Show how to write a function
Exponential functions
-an exponential function is in the form y=ab^x
-a cannot equal zero
-b is greater than zero
-and b cannot equal 1
-b is greater than 0
-a is the y-intercept of the graph.
-b is the stretch factor or the growth or decay factor
-an asymptote is the imaginary line that a graph approaches, but never reaches.
example...
(-1,25/3) (2,9/5)
(x,y) (x,y)
Plug the ordered pairs into the formula y=ab^x
25/3 = ab^-1 9/5 = ab^2
25/3 = a/b 9/(5b^2) =ab^2/b^2
25b/3 =a 9/5b62 = a
Since both equations are now equal to a, set the equations equal to eachother.
25b/3 = 9/5b^2
cross multiply
125b^3 = 27
divide by b and you are left with
b= 27/125 which simplifies to
b = 3/5
y = ab^x
y= 5(3/5)^x
The homework for this section is Page 426; 1-8; 10-31
Exponential Models lesson: 8-1
January, 2007
objuective: 1)to understnad how to write and graph exponential function.2.) use the function to model exponential growth and decay.
Exponential Form: Y=ab^x
(a can't equal 0), (b can't equal 0), and (b can't equal 1).
When b is greater than 1 its graph represents growth.
When b is less that 1 its grapg represents decay.
Example:
1.) y=3^x Growth
2.) Y= 1.5^x Growth
3.) Y=.33^x Decay
a, is always known as the y-intercept of the graph.
b, is known as the "stretch factor"
Steps for writing exponential functions when given two ordered pairs.
(4,8)(6,32) ordered pair
Y=ab^x
1.)plug first ordered pair into Y=ab^x
(cancel the b and solve equation for a)
2.) repeat step number 1 for the second ordered pair.
3.)set both equations (that are solved for a) equal to each other
4.) after you get what it is that a equals plug it into the formula and solve it for b.
5.) solve for exponential equation
**EXAMPLE**
(4,8)(6,32)
a.) 8=ab^4
8/b^4=ab^4/b^4
8/b^4=a
8=b^4
32=ab^6
32/b^6=ab^6/b^6
32/b^6=a
B.) 8/b^4=32/b^6
(b^6)8/b^4=32/b^6(b^6)
8b^6/b^4=32
8b^2=32
b^2=4
b=2
c.) 32/2^6=.5
Y=.5(2)^x
The form of the Exponentail Expression is y=ab^x
a cannot = 0 because the whole problem would be 0 and that would make a horizontal line on the x axis,
there are some restrictions on b b less 1,this means that b has to be greater than one becasue the graph would show growth,
0 less b greater1 this means that the graph would be between 0-1 and it would most likely be a fraction and this would show decay.
the x in the problem has to be a real number so that means there cannot be any negative,
asymtote are restrictions on the graph so that the graph cannot pass the x axis,
In lesson 8.1 we learn how to use exponential functions and logarithmic equations to help model and solve real life problems.
The objective in this lesson is to understand how to write an exponential function.An exponential function is an equation of the form Y=ab^x remember, a cannot equal 0, b>0, and b cannot equal 1.
A- will always be the y-intercept of the graph.
B- is known as the "stretch factor"
b>1= Growth, b<1= Decay
b==2 parent graph or basic graph
EXAMPLE:
(4,8)(6,32)
8=ab^4 32=ab^6
a=8/b^4 a=32/b^6
8/b^4=32/b^6
32b^4/b^4=8b^6/b^4
32=8b^2
b^2=4
b=2
a=8/2^4
a=1/2
y=1/2(2)^x
8-1: Exploring Exponential Models
*Exponential functions is an equation of the form y=ab^x,where a cannot equal 0, and b is greater than 0, and b cannot equal to 1.
-a function with general form y=ab^x where x is a real number, and a is not = to 0, b is greater than 0, and b is not 1.
*shortcuts for graphing y=ab^x
-a is always the y-int of the graph
-b is known as the "stretch factor"
*higher the b is,closer it is to the y-axis
*how to write an exponential function given two ordered pairs
1)Plug in
2)slove for a
3)set = and solve for b
4)plug in b in one of the two
equations and solve for a
5)write an exponential function in the form y=ab^x and plug in the numbers you found for a and b
EXAMPLE 1:
(4,8),(6,32)
8=ab^4 32=ab^6
8/b^4=a 32/b^6=a
(b^6)8/b^4=32/b^6(b^6)
8b^2=32
b^2=4
b=2
8/2^4=8/16= 1/2=a
*b is the growth factor,when b is greater than 1
*b is the decay factor,when 0greater than b less than 1
*asymptote- an imaginary line that a graph approaches, but never reaches, as x or y increases in absolute value
*exponential function x-axis (y=0) serves as the asymptote
Lesson 8.1a
For some data the best model is one that uses the independent variable as an exponent. An exponential function is a function with the form y=ab^x where x is a real number a must not equal zero and b isn’t equal to one.
To write and exponential function from two points you must solve for a from the formula y=ab^x
Write an exponential function y=ab^x for 2,2 and 3,4
2=ab² 4=ab³
a=2/b² a=4/b³
Set the two solutions equal to each other, and solve for b.
2/b² = 4/b³
Cross Multiply
2b³ = 4b²
Divide bothe sides by 2
b³ = 2b²
Divide bothe sides by b² to isolate b²
b = 2
Place b back into the solution for a to find a.
a= 2/2² a= 1/2
When graphing these exponential functions the line featured on the graph can be characterized as either a growth factor where b is greater than 1 and the line slopes up, or decay factor where b is less than 1 and the line slopes down.
The Aymptote is the line a graph nears as x or y increase in value.
8.1a:
Exponential Equations and Expressions
Exponential functions are equations written in the form of Y=ab^x. In such an equation "b" must be between 0 and 1 or greater than 1. B must not equal 1. For the function Y=ab^x 'b' will always equal the growth factor and 'a' will always equal the y-intercept. If 'b' is between 0 and 1 the grqph wil represent a decay if 'b' is greater than 1 it will represent growth.
To write an exponential equation for 2 ordered pairs solve both equations for 'a' i.e.
(2,2) (3,4)
2 = ab² --> 2/b² = ab²/b²
a = 2b
4 = ab³ --> 4/b³ = ab³/b³
a = 4/b³
Then set both equations equal to one another and solve for 'b'. When you've found by substitute it in one of the equations solved for a and solve it again. (incomplete)
Exploring Exponential Models
First you will need to know How to Write an Exponential Function.
*An Exponential Function is an equation of the form Y=ab^x.*
However, there are some rules:
- a cannot equal 0; b must be greater than 0; and b cannot equal 1.
SHORTCUT!
-for graphing-
*a will always be the y-intercept of the graph*
*b can be known as the "stretch factor"*
This guide may help:
-b=2 is the parent graph.
-It is a proper fraction if 0< b< 1.
-If y=(1/2)^x, it will be a decay.
-1< b< 2 is a wider graph than b=2.
*HINT*
If b is a decimal point or fraction between 0 and 1, it will be an exponential decay.
*The horizontal asymptote will always be y=0 in these graphs*
How do you write an expnonential function when you're given 2 Pairs?
1) write the 2 equations in the Y=ab^x form. (Using your x and y values given.) And solve for a.
2)Set the 2 a values equal to each other and solve for b.
3)Once you have your b value, substitute it in one of the a values.Solve for a.
4)Plug your a and b into the original Y=ab^x equation. YOU'RE DONE!! =)
Example:
(4,8) (6,32)
x y x y
8=ab^4 32=ab^6
8/b^4=a 32/b^6=a
8/b^4=32/b^6
b=2
8/2^4=> 1/2=a
y=1/2(2)^x
Good Luck!! =)
We started chapter 8, Exponential and Logarithmic Functions. In chapter 8 we will learn how to use exponential functions and logarithmic equations to model and solve real life problems. In lesson one, we learned the basics of exponential functions. We learned how to write an exponential function and how to sketch the graph of the function.
An exponential function is an equation in the form Y=ab^x. In this form, A cannot be zero, B has to be greater than zero, but not one. If B is a fraction, it will be a decay graph. If B is greater than one, it will be a growth.
Remember: A is always the Y-intercept of the problem, and B is known as the "stretch/growth factor." For example: Y=2(8)^x. 2 is the y-intercept (0,2) and 8 is the stretch factor.
We also learned how to write an exponential equation given two ordered pairs.
Steps for Writing the Equation Given Two Ordered Pairs:
1. Write two equations in the Y=ab^x. [use given ordered pairs to replace X and Y]
2. Solve for A.
3. Set both of the A equations equal to each other and solve for B.
4. Once you have your B, plug B back in to one of your A equations and solve for A.
5. Plug your A and B values back into the equation Y=ab^x.
Example: (1,2) and (2,3)
Step One: Replace X and Y using the given ordered pairs.
-- 2=ab^1 and 3=ab^2
Step Two: Solve for A.
--In the first equation divide both sides by b^1. So your answer is now: 2/b^1=a. In the second equation dived both sides by b^2 as well. So your answer is now: 3/b^2=a.
Step Three: Set both of your answers equal to each other and solve for B. So your equation should look like, 2/b^1=3/b^2. Now solve for B!
[Hint:] Multiply each side by the larger B term. In this case, you will multiply by b^2 on both sides.
Your answer should now be, 2b=3. Now, finish solving for B and you should get 3/2.
Step Four: Plug B back into one of your A equations and solve for A.
--3/b^2. 3/3/2^2. You should get 3/9/4. Now, for a fraction like this-Put your 3 over 1 and then multiply your outer and inner terms. So, it should look like this 3/1/9/4. So, multiply 3 and 4, and multiply 1 and 9. So, you should get 12/9. That reduces to, 4/3. So, A=4/3.
Step Five: Plug A and B back into your original equation.
-Your answer should look like this
Y=4/3(3/2)^x.
Good luck! (:
Lesson 8-1 Writing Exponential Functions and Sketching Exponential Graphs
First, to understand this lesson you have to know what an exponential equation is. An equation always uses an equal sign! (Equation=equal sign)
An exponential expression is just a way of expressing a number.
An exponential function is an equation in the form y=ab^x. That equation is important when graphing! There are restrictions, though. "a" cannot equal zero. Think about it...if "a" equaled 0 in the equation y=ab^x then the equation would automatically read y=0, which we all know is just the x-axis. Another restriction is "b" has to be greater then 0 and cannot equal one.
It is important to know that if "b" is the stretch factor or growth factor. "b" tells you how the slope will look. Will it be close to the y-axis or will it stretch away from the y-axis? It is also important to know that when "b" is between 0 and 1 the graph will have a decreasing slope (decaying graph). If "b" is greater than 1, then the graph will have an increasing slope (which indicates growth).
"a" is always the y-intercept of the graph.
Example: y=2(2)^x
a=2 so that means 2=y-intercept
b=2 since 2 is greater than one then the graph is "growing"
Example: y=1/2^x
a=1 anytime it is not present it equals one
b=1/2 since 1/2 is between 0 and 1 then the graph will be a decaying graph (sloping downwards)
*Horizontal asymptote is a boundary! It can be read as y=0.
Writing Exponential Functions
Here are the steps for writing exponential functions:
Step 1: Plug in the first ordered pair. Solve for "a".
Step 2: Plug in for the second ordered pair. Solve for "a".
Step 3: Set both equations that equal "a" equal to each other. Solve for "b".
Step 4: Plug in "b" to find "a" and then write the equation in y=ab^x form.
Example: (4,8), (6,32)
First thing you do is plug in y and x for the first ordered pair.
8=ab^4
Solve for "a".
a=8/b^4
Now you so he same thing for the second equation.
32=ab^6
Solve for "a".
a=32/b^6
Set both equations equal to each other.
8/b^4=32/b^6
Solve for "b".
b=2
Now go back to one of your "a" equations and plug in "b" to find "a".
a=8/2^4
a=1/2
Now all you have to do is plug "a" and "b" into the y-ab^x form and you're done!
y=1/2(2)^x
And that is your final answer!
Now we are through! Good Luck with lesson 8-1!
Chapter 8 - Using exponential functions and logarithmic equations.
Lesson Objectives-
(1) Obtain Knowledge of how to write and graph exponential functions.
(2) Apply the function to models of exponential growth and decay.
Part I
Vocabulary
-Exponential Function- A function in the form y=ab^x in which a does not equal, 0 b is greater than 0, and b is not equal to 1.
-Growth Factor- An exponential function in which b is greater than 1.
-Decay Factor- An exponential function in which b is less than 1 but greater than 0.
-Asymptote- A line that a graph approaches as x or y increases in absolute value.
Author Note: Lesson 8-1 not complete but to be continued in part II.
"8.1 a Lesson Summary"
In this lesson you will be able to use exponential funtions & logarithmic equations to model and solve real life problems. The first thing you want to know how to do is how to write an exponential function. The exponential function Y=ab^x. There are some restrictions to the a and the b. "A" can not equal 0. The b has to be positive or greater than 0. B can not equal 1. B is known as the stretch factor. In other words it makes the bar stretch up the graph. The a will always be the y-intercept. To determine if the graph has exponential growth or exponential decay is easy. If the b is greater than 1 and the a is greater than 0 then it has exponential growth. If the b is below 1 though then it has exponential decay(0 < b < 1). To write two equations in form y=ab^x you first must solve both equations for a, set them both equal to one another, & then solve for b. Substitute the value b into one of the two equations from step 2. The write your answers in the y=ab^x form. For example: (4,8)(6,32) y=ab^x write 8=ab^4. Then solve for a. So your'e going to divide b^4 to the 8 so it will look like this 8 over b^4 = a then do the other ordered pair. 32=ab^6, then make it 32 over b^6 . Okay now that you got both a's ... set them equal to each other. Then solve for b the same way you solved for a . Once you have solved for b and got your answer plug it into one of your a's. I'm going to put my b(2) into my one of my a's. Im going to use the a 8 over b^4 . So now it will be 8 over 2^4. 2^4 = 16 so 8 over 16 equals .5 or 1/2. Now write your exponential equation. Y=.5(2)^x and your done. Its that simple. Now on to graphing these exponential equations. Remember the word asymptote. Its an imaginary line that a graph approaches but never reaches as x or y increases in absolute value. The x-axis serves as the asymptote for exponential equations.
In lesson 8.1, we lwarn to write and graph exponential functions, also finding the exponential growth and decay.
REMEMBER: anything raised to the zero power=1
An exponential function is an equation of the form.
Y=AB^X
A cannot=0, B > 0 and B cannot=1 or be a negative.
shortcuts for graphing Y=AB^X
A is always the Y-intercept fo the graph.
B is known as the stretch factor
B=2, "Parent Graph"
if 0 < B < 1, i.e B is the power
EXAMPLE:
(2,2) + (3,4)"plug in solve for A"
2/B=AB^2/B
2/B^2=A
4/B^3=AB^3/B^3
4/B^3=A
"set = solve for B"
2/B^2=4/B^3
2B^3/B^2=4B^2/B^2
2B/2=4/2
B=2
Chapter 8.1
Exploring Exponential Models.
Vocabulary:
An Exponential Function is a function with general form y=ab^x where x is a real number.
Asympote-an imaginary line that a graph approaches, but it never reaches, as x or y increases in abosolute value.
The x-axis (y=0)- serves as the asympote for the exponential equation.
To write an exponential function use the form y=ab^x.
a can not be equal to 0,b has to be greater than 0 and b can not equal 1.
The y-intercept of the graph will be a and b is the growth or also known as the "stretch factor".
If the graph increases upward it is the growth factor and if it increases down it's the decay factor.
To write an exponential function when given two ordered pairs, first write the form in y=ab^x and replace both y and x by the given ordered pairs in each equation. Solve both equations for a and set them equal to one another and then solve for b. Then substitute the value of b into one of the two equations from the previous two equations after that solve for a.
Write the final equation by using the values founded for a and b.
Example: (2,2)&( 3,4).
2=ab^2 a=2/b^2
~Solve for a~
4=ab^3 divide by b^3
~Set equal and solve for b~
2/b^2=4/b^3 * cross multiply and divide-2b^3/b^2 = 4b^2/b^2
2b=4 b= 2
a=1/2
~*Substitute the values of a&b*~
*Final Answer*
y=1/2(2)^x
Yesterday in lesson, 8.1a, we learned how to use eponential functions to model and solve real life problems.
objective:understand how to write and graph an expotential function. We also learned to use exponential functions to show growth and decay.
exponential function: an equationof the form y=ab^x
Ex:y=3(5)^x in this equation there would be an exponential growth because 5 or (b) is greater than 1 (0 is less than band b and b is less than or greater than 1) the 3 or (a) determines where the line crosses the y-axis also remember for these problems the horizontal asymtote is y=0 (or the x-axis)(a can't equal 0)
The other thing we learned was to make an exponontial function useing to points.
(2,2)(3,4)
use y=ab^x
Step 1:solve for a in both points
Step 2:set a1 & a2 equal to each otherand solve for b
step 3:choose an equation for a, plug in b and solve for a
Step 4:plug answers for a and b into the equation y=ab^x
Our class has learned how to model exponential growth and decay. Some real world examples that model growth and decay are the following: radiocarbon dating, and depreciation/appreciation of cars. The needed function is y=ab*.x is a real number, a cannot equal 0, and b has to be greater than 0 but not equal to 1. To determine whether the function models growth or decay, the b value should be examined. When b is greater than 1, the graph models growth, when b is less than 1, the graph models decay. a is the y-intercept, and b is the growth or decay factor. For the parent function, y=2*, a is an understood 1, and the estimated curve of the graph is about 30 degrees away from the y-axis. To graph the function y=4*, you can either make a table of values then plot the points, or simply sketch the graph by eye, drawing a smooth curve in the direction the arrow show go. (FYI: as b gets larger, the closer the graph will be to the y-axis. As b gets smaller, the graph will be futher away from the y-axis.)If you are not given the b value, and you know the rate of interest, you can find b by using the equation b=1+r. Ex: rate is 1.36%. substitute 0.0136 into the equation for r. This will give you the value of b, then you simply plug b into the y=ab* equation. There will also be times when you will be asked to write an exponential function given two ordered pairs. Ex: (2,4) (8,2). To solve these problems, substite the x and y values into two seperate y=ab* equations. 4=ab^2 and 2=ab^8. Solve each equation for a(a=4/b^2 and a=2/b^8).Set the two a problems equal to each other (4/b^2=2/b^8). Solve the equation for b. After you find b, substitute b into one of the two a equations. Then solve for a. Write your answer in the equation y=ab*, substituting the a and b values.
Lesson 8.1 is about graphing and ssolving exponential functions. The form of a function is y=ab×, where a cannot equal 0 and b cannot equal 1. In this form, a represents the y-intercept, and b is the "stretch factor", or how much the graph stretches out. If b is between 0 and 1, the graph decays. If b is greater than 1, the graph grows or increases.
-incomplete, will finish later.
In the lesson 8.1 we learned about Exponential Equations (ex. 2^2=1.9^2) and Exponential Expressions (ex.3^x+2).
In this lesson we found out that y=ab^x is exponential function form. A few rules for that function are: a cannot equal 0; bgreater than0; b cannot equal 1.
When graphing y=ab^x, the graph becomes either an exponential growth or exponential decay. In an expontial growth the bgreater than1. While in an exponential decay, b is 0 less than b less than1.
***Shortcuts
a= y intercept
b=stretch factor
If b=2 is known as Parent Graph.
If 0less thanbless than1; i.e. b is a proper fraction.
Ex. 4(2)^x
a=4 b=2 y int.=4
The next thing we learned is how to a write exponential function.
The problem starts out with two different coordinates.
1. Write 2 equations in form of y=ab^x. Use ordered pairs to replace x and y for each equation.
2.Solve both equations for a set, then equal them to each other, and solve for b.
3.Substitute the value of b into one of the equations from step 2; then solve for a.
4.Write final equation by using the values you found for a and b.
Examples*
(2,2),(3,4)
y=ab^x
2=ab^2 4=ab^3
2/b^2=a 4/b^3=a
2/b^2=4/b^3
2b^3/b^2=4b^3/b^3
2b=4
b=2
Plug 2 into 1st equation.
2/b^2 b=2
2/2^2 = 2/4
a=1/2
Answer y=1/2(2)^x
The last and final thing we learned is how to identify the horizontal asymptote. An asymptote is a line that a graph approaches as x or y increases in absolute value.
-Lauren Thornton
Chapter 8.1
Part II
Examples:
(1) Graphing Exponential Growth
As learned from the vocabulary in part I we know that b must be greater than 1 in order to be growth.
In this example we have y=2^x.
A way to go about starting this equation is by making a table of values. Pick a point on the graph and substitue it for x.
Table
---------------------------
X Y
-3 1/8
-2 1/4
-1 1/2
0 1
1 2
2 4
3 8
That is an example table of values given for this equation.
After you graph you may notice that the y-intercept is (0,a). The abscents of an a tells us that it is an understood 1 so the y-intercept is (0,1).
(2) Modeling population
If you know the rate you can model population by b=1+r. Fill in the rate of growth for r and then solve. After you solve for b plug b into the equation y=ab^x and do the same as in step 1.
(3) Creating an Exponential Function
If given a pair of two ordered pairs and asked to graph them here are the steps for solving the equation.
First substitute the first ordered pair into the equation y=ab^x then solve for a. Do that again with the next ordered pair. After that set both sides equal to each other and solve for b. Once you've gained b plug it into one of the original problems for your solution.
This session ends part II.
Ok, I hope I am doing this right.
Chapter 8 Lesson 1 discusses the general uses of exponential functions. Also, it shows how to correctly write and graph these functions.
Exponential Func.Equation- y=ab^x
* b Cannot = 0 or 1, b>0
* a is your y-intercept
* B is your "stretch". The higher the b-value the closer to the y-axis. The lower the b value the closer to the x-axiz
If your b value is less then 1 your graph will fall left to right gradually decending. If your b value is greater then 1, it will will be growing form left to right.
EX: (4,8) (6,32)
1)Plug in 1st set of values
8=ab^4
2)Solve for A
8/b^4=a
3) PLug in 2nd amount along with a
32=(8/b^4)(b^6)
4) divide b's
32=8b^2
5)Solve
4=b^2; b=2
**** y= 1/2(2)^x
Hello
In ch. 8.1 the objective is to use exponential functions and logarithmic equations to model and solve real life problems
An exponential function is written as y=ab^x where x is a real number, a doesn't equal 0, b > 0, and b doesn't equal 1.
When b > 1, b is the growth factor, and when 0 < b < 1, b is a decay factor.
An asymptote is an imaginary line that a graph approaches, but never reaches, as x or y increases in absolute value
The x-axis (y=0) serves as the asymptote for exponential equations
Shortcuts for graphing y=ab^x
a is always the y-intercept of the graph
b is known as the growth or decay factor
(4,8) and (6,32)
solve the equations for a.
8=ab^4 and 32=ab^6
8/b^4=a and 32/b^6=a
make the equations equal to each other and solve for b.
8/b^4=32/b^6
8b^6=32b^4
8b^6/b^4=32b^4/b^4
8b^2/8=32/8
b^2=4
√b^2=2
Plug b into one of the original problems and solve for a.
8/2^4=a
8/16=a
1/2=a
put answer into the y=ab^x form
y= .5(2)^x
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