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Slides: 28
Duration: 00:14:37
Filename: C:\Users\jpage\Documents\NCVPS Learning Objects\Math 3\Math 3 Exponentials and Logs Navigation to PPT
W\Mod 6 Lesson 2 Notes.ppt
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Slide 1
Graphing Exponential Functions,
Logarithms, and Logarithmic
Functions
Notes:
Module 6 Lesson 2
Graphing Exponential Functions, Logarithmic &
Logarithmic Functions
Duration: 00:00:18
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Slide 2
What does an Exponential Function
look like?
Duration: 00:00:22
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Slide 3
Whar are Exponential Functions used
for?
Duration: 00:00:18
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Notes:
What does an exponential function look like?
Recall that a linear function looks like a line. For
example f(x) = mx + b
The basic exponential function simply looks like
this (look at the picture below): f(x) = b^x (it just
has a variable as an exponent)
Notes:
What are exponential functions used for?
An exponential function is used to show growth
or decay that change at a faster and faster rate.
Some examples are compound interest,
uninhibited population growth, and the decay of
radioactive material.
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Slide 4
f(x) = b^x
Duration: 00:00:46
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Slide 5
Asymptotes
Duration: 00:00:18
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Notes:
For the graph of f(x) = b^x
1) The graph will always contain the point
(0, 1)
2) When b > 1, the graph will rise from left
to right.
3) When 0 < b < 1, the graph will fall from
left to right.
4) The graph in either case above will
always be in quadrants I & II, starting in
II.
5) The graph will approach the x-axis, but
will never touch the x-axis (therefore the
x-axis is called an asymptote)
6) The domain is always all real numbers
7) The range is always y > 0.
Notes:
Asymptote – A line that graph approaches (but
never touches) as it moves away from the origin.
Here, the asymptote is y = 3 because the graph
will get closer and closer to y = 3, but it never
actually makes it to y =3.
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Slide 6
f(x) = b^x properties
Duration: 00:00:36
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Notes:
With the graph on the left, notice that as x
increases, y increases. This is true for f(x) = b^2
when b > 1.
With the graph on the right, notice that as b
increases the graph becomes steeper (grows
closer to the y-axis).
Another way to look at it is that as b increases
the faster the graph rises.
Notice, the asymptote for all of these graphs is y
= 0 because the graphs all get closer and closer
to y = 0 but they never actually make it to y = 0.
Slide 7
Graphing Exponential Functions
Duration: 00:00:27
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Notes:
To graph an exponential function create a table.
Be sure to find several ordered pairs, enough to
see how rapidly the graph grows, and then
connect the points with a line.
Also notice that the graph is very close to the xaxis, BUT it NEVER touches the x-axis.
Since it never touches the axis, it just continues
to get closer and closer, the x-axis is called an
asymptote.
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Slide 8
f(x) = b^x when b > 1
Duration: 00:00:50
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Notes:
Case 1: b>1
Take a look at the value of y = 2^x as x changes.
In your calculator, find the value of y when x = 2,
3, 4, and 5. (graph the function, then look at the
table)
What is happening to the value of y?
Then, find the value of y when x = -1, -2, -3, and
-4. (look at the table)
What is happening to the value of y?
You should see that as x decreases, the y value
approaches 0 and as x increases the y value
gets large very quickly.
As x increases, y increases. As x decreases, y
decreases, but y is never negative!
Look at the next slide . . .
Slide 9
f(x) = b^x when b > 1
Duration: 00:00:38
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Notes:
y ^x, b > 1
The graph passes through the points (0, 1) and
(1, m)
The domain of the function is all real numbers
The range of the function is all real numbers that
are strictly greater than 0 (y >0), i.e. all positive
numbers
The function is increasing as x increases
There is a horizontal asymptote at y = 0 (the xaxis). The function approaches the line y = 0 as
x decreases.
In other words, the value of the function
approaches 0 as x decreases.
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Slide 10
f(x) = b^x when 0 < b < 1
Duration: 00:00:46
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Notes:
Case 2: 0 < b < 1
Take a look at the value of y = (1/2)^x as x
changes.
In your calculator, find the value of y when x = 2,
3, 4 and 5. (graph the function, then look at the
table)
What is happening to the value of y?
Then, find the value of y when x = -1, -2, -3, and
-4.
What is happening to the value of y?
As x increases, y decreases. As x decreases, y
increases, but y is never negative!
The value of y is approaching 0 as x increases.
Look at the next slide . . .
Slide 11
f(x) = b^x when 0 < b < 1
Duration: 00:00:39
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Notes:
y = b^x, 0 < b < 1
The graph passes through the points (0, 1) and
(1, m).
The domain of the function is all real numbers.
The range of the function is all real numbers that
are strictly greater than 0 (y > 0), i.e. all positive
numbers.
The function is decreasing as x increases.
There is a horizontal asymptote at y = 0 (the xaxis). The function approaches the line y = 0 as
x increases. In other words, the value of the
function approaches 0 as x increases.
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Slide 12
e^x
Duration: 00:00:38
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Notes:
A special case f(x) = b^x: The Natural
Exponential Function ~ f(x) = e^x
e is a very important and frequently used
irrational number whose value is approximately
2.71828
If you look at your calculator, the “e” button is the
nd
2 function of the division sign.
If you want to raise the number e to a power, you
nd
can use the “e^x” button, which is the 2 function
of the “LN” button, on the left side of your
calculator.
The number e is often used as the base of an
exponential function.
Slide 13
e^x
Duration: 00:00:22
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Notes:
Natural Base e (Euler number) – An irrational
number defined as:
As n approaches positive infinity, the value of (1
+ 1/n)^n approaches e ~ 2.718281828459
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Slide 14
Graph of e^x
Duration: 00:00:06
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Slide 15
Activity
Duration: 00:00:38
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Notes:
The graph of the function is show below: y = e^x
Notes:
Activity: The number e is actually the number
that the quantity (1 + 1/n)^n approaches as n
gets larger and larger . . .
Write the following table on a piece of paper, and
fill it in using the values you find from the
calculator.
Plug the appropriate values for n into the
expression (1 + 1/n)^n, and calculate the value of
the expression using your calculator.
(I’ve done the first couple for you . . . )
What do you notice about the value of (1 +
1/n)^n as n gets bigger?
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Slide 16
e^x
Notes:
Duration: 00:00:10
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Slide 17
Transformations
Duration: 00:00:48
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Notes:
Transformations
Vertical Shifts:
The graph of the function is shifted up or down a
certain number of units.
x
x
For f(x) = b , f(x) = b + c shifts the graph up.
x
x
x
x-k
x
x+k
For f(x) = b , f(x) = b - c shifts the graph down.
Horizontal Shifts:
The graph of the function is shifted left or right a
certain number of units.
For f(x) = b , f(x) = b shifts the graph right.
(opposite of what you would think)
For f(x) = b , f(x) = b
shifts the graph left.
(opposite of what you would think)
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Slide 18
Transformations
Duration: 00:00:26
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Notes:
Transformations
Reflection of the graph:
The graph of the function is reflected over an
axis:
x
x
x
-x
For f(x) = b , f(x) = -b reflects the graph about
the x-axis.
For f(x) = b , f(x) = b reflects the graph about
the y-axis.
Slide 19
Transformations
Duration: 00:00:53
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Notes:
Transformations
Vertical Compressions/Stretches:
The graph of the function is compressed or
stretched vertically.
x
x
x
x
For f(x) = b , f(x) = a*b where a > 1 gives a
vertical stretch .
For f(x) = b , f(x) = a*b where 0< a < 1 gives a
vertical compression .
Horizontal Compressions/Stretches:
The graph of the function is compressed or
stretched horizontally.
x
ax
For f(x) = b , f(x) = b where a > 1 gives a
horizontal stretch .
x
ax
For f(x) = b , f(x) = b where 0< a < 1 gives a
horizontal compression .
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Slide 20
Activity
Duration: 00:00:52
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Notes:
Graph y = 3^(x + 2) + 1 and describe its
characteristics. (Notice the parent graph is f(x) =
3^x )
The 2 moves the graph 2 unites left ad the 1
moves the graph 1 unit up.
The graphs are both increasing.
The graphs both have asymptotes, however f(x)
= 3^x has an asymptote of y = 0 and f(x) = 3^(x +
2) + 1 has an asymptote of y = 1
The domain of both graphs is all real numbers.
The range of f(x) = 3^x is y > 0
The range of f(x) = 3^(x + 2) + 1 is y > 1
Slide 21
Finding the Equation of an
Exponential Function
Notes:
Finding the Equation of an Exponential Graph
Duration: 00:00:04
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Slide 22
Finding the Equation of an
Exponential Function
Duration: 00:00:51
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Notes:
Find the equation of the exponential function.
1) The asymptote is at y = -1 (which tells you
there is a vertical shift down 1)
So the function must look something like this: y
= ab^x - 1
2) The y-intercept is (0,3)
So substitute the y-intercept, y = 3 and x=0, into
the equation above, y = ab^x - 1, to find a.
y = ab^x – 1
3 = ab^0 – 1
3=a–1
3=a–1
a=4
3) Plugging in the values, you now have the
equation: y = 4b^x - 1
Slide 23
Finding the Equation of an
Exponential Function
Duration: 00:00:57
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Notes:
4) Pick a second point. This graph has an
x-intercept (-2, 0) due to the shift. Use
this to find b.
y = 4b^x – 1
0 = 4b^(-2) – 1
1 = 4b^(-2)
¼ = b^(-2)
¼ = 1/b^2
b=2
5) Now you have y = 4(2^x) – 1 Graph this in
your calculator to check your answer.
Note: Not all modeling problems are as clear
and straightforward as this one. Sometimes
you will have to estimate the x – or y –
intercepts as well as the base function.
Modeling data can get tricky and it is
important to remind yourself to be patient
and try different values for a, b and c in your
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base function to find the appropriate model.
Slide 24
Example
Duration: 00:00:55
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Notes:
Write an exponential function for the graph that
contains the following 2 points:
(0, 8) and (4, 19208)
1) Using the y-intercept: (0, 8)
y = ab^x
8 = ab^0
8 = a(1)
8=a
This gives you y = 8b^x
2) Using the second point: (4, 19208)
y = 8b^x
19208 = 8b^4
2401 = b^4
th
th
4 root of 2401 = 4 root of b^4
7=b
Final answer: y = 8(7^x)
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Slide 25
Logarithms
Duration: 00:00:37
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Slide 26
Logarithms
Duration: 00:00:12
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Notes:
Logarithms
Definitions: The logarithmic function with base b,
where b > 0, (b not equal to 1), is written y = log
(b) x, and is defined as y = log (b) x if and only if
x = b^y. (The logarithm of base 10 is known as
the common logarithm, and is written y = log
(10)x = log x. If no base is provided, then it is an
assumed base of 10.
Notes:
Logarithms
Logs are inverses of exponentials
The graph of a log would be the reflection of the
exponential graph across the diagonal line y = x.
Slide 27
Graphs of logarithmic functions
Duration: 00:00:04
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Notes:
Graphs of logarithmic functions.
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Slide 28
Graphs
Duration: 00:00:06
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Notes:
Please visit these sites for more information on
exponential and logarithmic graphs.
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