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Properties of Exponential Functions

Properties of Exponential Functions. 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… Evaluate and simply expressions containing exponents

caleb-stokes
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Properties of Exponential Functions

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  1. Properties of Exponential Functions In Grade 11 and 12 College/University Math

  2. The 3 Overall Expectations Simply put, the grade 11/12 curriculum asks that the students be able to… • Evaluate and simply expressions containing exponents • Make the connection between the numeric, graphical, and algebraic representations (Graph them! Transform them!) • Solve real-world applications involving exponential functions.

  3. 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!

  4. The Exponent Laws

  5. This way.. • This way, students can simply algebraic expressions containing integer and rational exponents… • Examples: simplify the following two • 41/2 x 4 ½ = • X3 / X1/2= • (X6y3)1/3=

  6. 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.

  7. 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

  8. 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

  9. 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.

  10. 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

  11. 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

  12. Real Life Questions: Exponential Decay

  13. 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?

  14. Moving Further into the Realms of Functions… LOGARITHMS!

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