# Properties of Exponential Functions ```In Grade 11 and 12 College/University Math
The 3 Overall
Expectations
students be able to…
1. Evaluate and simply expressions containing
exponents
2. Make the connection between the numeric,
graphical, and algebraic representations (Graph
them! Transform them!)
3. Solve real-world applications involving exponential
functions.
How
to
Get
Started…
Here are some functions
that the students should
be familiar with after
learning Trigonometric
functions…
Hint: this picture is a
warmup of what’s to
come!
The Exponent Laws
Law
Example
x1 = x
61 = 6
x0 = 1
70 = 1
x-1 = 1/x
4-1 = 1/4
xmxn = xm+n
x2x3 = x2+3 = x5
xm/xn = xm-n
x6/x2 = x6-2 = x4
(xm)n = xmn
(x2)3 = x2&times;3 = x6
(xy)n = xnyn
(xy)3 = x3y3
(x/y)n = xn/yn
(x/y)2 = x2 / y2
x-n = 1/xn
x-3 = 1/x3
And the law about Fractional Exponents:
This way..
 This way, students can simply algebraic expressions
containing integer and rational exponents…
 Examples: simplify the following two
 41/2 x 4 &frac12; =
 X3 / X1/2=
 (X6y3)1/3=
So then, Introducing Exponential Functions!
 They involve exponents
Examples: y=2x y=3x y=bx
 Start off with f(x) = bx
 x is the exponent
 b is the base
Students should be able to graph with calculators, paper and pencils, and
graphing technology based on a table of values.
Then looking at a basic exponential
function, students need to…
1.4 – determine the key
properties relating to domain
and range, intercepts,
increasing/decreasing intervals,
and asymptotes.
f(x)=2x
And the properties are…
 Domain:
 Range:
The set of real numbers
Set of positive real numbers
 Intercepts:
Dependant on the a value of f(x)=abx
 Increasing/Decreasing Interval:
Increase if b&gt;1,
Decrease if b&lt;1
 Asymptotes:
Horizontal asymptote on x=0
x
f(x)=e
 Although there is no instruction to teach the function
f(x)=ex, it would be useful to introduce the base e.
 The numerical value of e is approximately 2.71828183
 Later on, this will be expanded in logarithmic
functions.
The transformations!
 Students are to investigate, using technology, the roles
of the parameters a, k, c, and d in functions of the
form f(x) = a ek (x - c) + d, and compare it to the graph
of f(x)=ax
It may be helpful for the visual learners to use this
interactive script online to see the patterns. (However,
this pattern rebounds off the original graph of f(x)=ex )
http://archives.math.utk.edu/visual.calculus/0/shifting.5/index.html
Approximation Activity
Get into groups and, using your body, demonstrate the
two graphs below and then describe the transformation
involved from f(x)=3x to f(x)=0.3x-2-5
Real Life Questions:
Exponential Decay
Year
Population
1983
125
1984
75
1985
50
1986
37
1987
32
1988
25
1989
22
1990
20
1991
18
1992
16
1993
14
1994
10.5
1995
10
First Differences
Ratio
Exponential Decay –
Computers continued…
Neatly sketch a graph of the data from the table on the previous
page. When choosing your scale for the horizontal axis,
consider question 4 below. After you have plotted the points,
draw a smooth curve through them.
 Using the graph, comment on the shape of the curve. Use
words such as the following in your description: increasing,
decreasing, quickly, slowly.
 Use your graph to predict the number of students per computer
in the year 2006.
 Is the answer from question #4 surprising? Why or why not?
Moving Further into the Realms of
Functions…
LOGARITHMS!
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