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In Grade 11 and 12 College/University Math
The 3 Overall Expectations
Simply put, the grade 11/12 curriculum asks that the 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 LawsLaw 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 = x2×3 = 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 two41/2 x 4 ½ =X3 / X1/2=
(X6y3)1/3=
So then, Introducing Exponential Functions!They involve exponentsExamples: y=2x y=3x y=bx
Start off with f(x) = bx
x is the exponentb 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:Intercepts:Increasing/Decreasing Interval:
Asymptotes:
The set of real numbers
Set of positive real numbers
Dependant on the a value of f(x)=abx
Increase if b>1,Decrease if b<1
Horizontal asymptote on x=0
f(x)=ex
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
First Differences Ratio1983 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
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!