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exponential functions

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They are different than any of the other types of functions we ve studied ... the graphs 2x, 3x , and 4x ... We have graphed and its inverse y = x Reflected over ... – PowerPoint PPT presentation

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Title: exponential functions


1
exponential functions
2
Weve looked at linear and quadratic functions,
polynomial functions and rational functions. We
are now going to study a new function called
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
3
Compare the graphs 2x, 3x , and 4x
Characteristics about the Graph of an Exponential
Function where a gt 1
  1. Domain is all real number

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 except for 0 lt
a lt 1. The function is decreasing.
6. The x-axis (where y 0) is a horizontal
asymptote for x ? - ?
4
Equations with x and y Interchanged
Graph
Graph
y x
We have graphed
and its inverse
5
Reflected over 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 (well see
some of these later this chapter). To model
these we need to learn about a special base.
6
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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