exponential functions

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exponential functions
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Lets examine exponential functions. They are
different than any of the other types of
functions weve studied because the independent
variable is in the exponent.
Lets look at the graph of this function by
plotting some points.
x 2x
3 8
2 4
BASE
1 2
0 1
Recall what a negative exponent means
-1 1/2
-2 1/4
-3 1/8
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Compare the graphs 2x, 3x , and 4x
Characteristics about the Graph of an Exponential
Function where a gt 1
1. Domain is all real numbers
2. Range is positive real numbers
3. There are no x intercepts because there is no
x value that you can put in the function to make
it 0
What is the domain of an exponential function?
What is the range of an exponential function?
What is the x intercept of these exponential
functions?
Can you see the horizontal asymptote for these
functions?
What is the y intercept of these exponential
functions?
Are these exponential functions increasing or
decreasing?
4. The y intercept is always (0,1) because a 0
1
5. The graph is always increasing
6. The x-axis (where y 0) is a horizontal
asymptote for x ? - ?
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All of the transformations that you learned apply
to all functions, so what would the graph of
look like?
up 3
down 1
right 2
Reflected over x axis
up 1
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Reflected about y-axis
This equation could be rewritten in a different
form
So if the base of our exponential function is
between 0 and 1 (which will be a fraction), the
graph will be decreasing. It will have the same
domain, range, intercepts, and asymptote.
There are many occurrences in nature that can be
modeled with an exponential function. To model
these we need to learn about a special base.
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The Base e (also called the natural base)
To model things in nature, well need a base that
turns out to be between 2 and 3. Your calculator
knows this base. Ask your calculator to find e1.
You do this by using the ex button (generally
youll need to hit the 2nd or yellow button first
to get it depending on the calculator). After
hitting the ex, you then enter the exponent you
want (in this case 1) and push or enter. If
you have a scientific calculator that doesnt
graph you may have to enter the 1 before hitting
the ex. You should get 2.718281828
Example for TI-83
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If au av, then u v
This says that if we have exponential functions
in equations and we can write both sides of the
equation using the same base, we know the
exponents are equal.
The left hand side is 2 to the something. Can we
re-write the right hand side as 2 to the
something?
Now we use the property above. The bases are
both 2 so the exponents must be equal.
We did not cancel the 2s, We just used the
property and equated the exponents.
You could solve this for x now.
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The left hand side is 4 to the something but the
right hand side cant be written as 4 to the
something (using integer exponents)
Lets try one more
We could however re-write both the left and right
hand sides as 2 to the something.
So now that each side is written with the same
base we know the exponents must be equal.
Check
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au