Exponential Functions. Our presentation today will consists of two sections. Our presentation today...
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Transcript of Exponential Functions. Our presentation today will consists of two sections. Our presentation today...
![Page 1: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/1.jpg)
Exponential Functions
An is a function
of the form ,
where 0, 0, and 1,
and t
expone
exponent vahe riab
ntial f
must be a .
unction
le
xy
b
b
b
a
a
![Page 2: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/2.jpg)
Our presentation today will consists
of two sections.
Our presentation today will consists
of two sections.
Section 1: Exploration of exponential functions and their graphs.
Section 2: Discussion of the equality property and its applications.
![Page 3: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/3.jpg)
First, let’s take a look at an exponential function
2xy x y
0 1
1 2
2 4
-1 1/2
-2 1/4
![Page 4: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/4.jpg)
So our general form is simple enough.
The general shape of our graph will be determined by the exponential
variable.Which leads us to ask what
role does the ‘a’ and the base ‘b’ play here. Let’s take a
look.
xy a b
![Page 5: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/5.jpg)
First let’s change the base b
to positive values
What conclusion canwe draw ?
![Page 6: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/6.jpg)
Next, observe what happens when b assumes a value such that
0<b<1.
Can you explain whythis happens ?
![Page 7: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/7.jpg)
What do you think will happen if ‘b’ is
negative ?
![Page 8: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/8.jpg)
Don’t forget our definition !
Can you explain why ‘b’ is restricted from assuming negative
values ?
Any equation of the form:
y a b x , where a 0, b 0 and b 1
![Page 9: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/9.jpg)
To see what impact ‘a’ has on our graph
we will fix the value of ‘b’ at 3.
What does a largervalue of ‘a’ accomplish ?
xy a b
![Page 10: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/10.jpg)
Shall we speculate as to what happens when ‘a’ assumes
negative values ?
Let’s see if you are correct !
![Page 11: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/11.jpg)
0, 0 and 1xy a b where a b b
![Page 12: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/12.jpg)
Our general exponential form is “b” is the base of the function and
changes here will result in: When b>1, a steep increase in the
value of ‘y’ as ‘x’ increases. When 0<b<1, a steep decrease in the
value of ‘y’ as ‘x’ increases.
y a b x
![Page 13: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/13.jpg)
We also discovered that changes in “a” would change the y-intercept on its corresponding graph.
Now let’s turn our attention to a useful property of exponential functions.
xy a b
![Page 14: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/14.jpg)
The Equality Property of Exponential Functions
Section 2
![Page 15: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/15.jpg)
We know that in exponential functions the exponent is a variable.
When we wish to solve for that variable we have two approaches we can take.
One approach is to use a logarithm. We will learn about these in a later lesson.
The second is to make use of the EqualityProperty for Exponential Functions.
![Page 16: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/16.jpg)
Suppose b is a positive number other
than 1. Then b x1 b x 2 if and only if
x1 x2 .
The Equality Property for ExponentialFunctions
This property gives us a technique to solve equations involving exponential functions. Let’s look at some examples.
Basically, this states that if the bases are the same, then we can simply set the exponents equal.
This property is quite useful when we are trying to solve equations
involving exponential functions.
Let’s try a few examples to see how it works.
Basically, this states that if the bases are the same, then we can simply set the exponents equal.
This property is quite useful when we are trying to solve equations
involving exponential functions.
Let’s try a few examples to see how it works.
![Page 17: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/17.jpg)
Example 1:
32x 5 3x 3(Since the bases are the same wesimply set the exponents equal.)
2x 5 x 3x 5 3
x 8
Here is another example for you to try:
Example 1a:
23x 1 21
3x 5
![Page 18: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/18.jpg)
The next problem is what to do when the bases are not the
same.
32x 3 27x 1
Does anyone have an idea how
we might approach this?
![Page 19: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/19.jpg)
Our strategy here is to rewrite the bases so that they are both
the same.Here for example, we know that
33 27
![Page 20: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/20.jpg)
Example 2: (Let’s solve it now)
32x 3 27x 1
32x 3 33(x 1) (our bases are now the sameso simply set the exponents
equal)2x 3 3(x 1)
2x 3 3x 3
x 3 3
x 6
x 6
Let’s try another one of these.
![Page 21: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/21.jpg)
Example 3
16x 1 1
32
24(x 1) 2 5
4(x 1) 54x 4 5
4x 9
x 9
4
Remember a negative exponent is simply another way of writing a fraction
The bases are now the sameso set the exponents equal.
![Page 22: Exponential Functions. Our presentation today will consists of two sections. Our presentation today will consists of two sections. Section 1: Exploration.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3a5503460f94b2bf26/html5/thumbnails/22.jpg)
By now you can see that the equality property is
actually quite useful in solving these problems.
Here are a few more examples for you to try.
Example 4: 32x 1 1
9
Example 5: 4 x 3 82x 1