2. 5/16/14 OBJ: SWBAT graph and recognize exponential functions. Bell Ringer: Start notes for Exponential functions Homework Requests: pg 246 #1-29 odds 37, 39, 41, 43 Homework: p286 #1-19 odds Read Sect. 3.2 Announcements: Quiz next Week Worksheet for over the weekend. Maximize Academic Potential Turn UP!

3. 5/16/14 Obj: SWBAT solve exponential equations Bell Ringer: Go over Quiz; Turn In project HW Requests: Unit circle WS pg 743 #11-21 odds, 23-32 Homework: Complete skills practice Read Section 10.1 Ex 1-3 Announcements: Law of Cosines/Sines Project due today Friday 5/16 Extended Tues. Must do Ch 13 test thurs Education is Power! Education is Power You can DO IT!

4. Exponential Functions constant a is the initial value of f(x) at x = 0, b is the base

5. Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent. Let’s look at the graph of this function by plotting some points. x 2 x 3 8 2 4 1 2 0 1 -1 1/2 -2 1/4 -3 1/8 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 Recall what a negative exponent means: BASE

6. a> 0, b > 1 exponential growth, 0<b< 1 Exponential Decay Pg 280

7. 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. If a u = a v, then u = v 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 2’s, We just used the property and equated the exponents. You could solve this for x now. The Equality Property for Exponential Functions

8. Let’s try one more: The left hand side is 4 to the something but the right hand side can’t be written as 4 to the something (using integer exponents) We could however re-write both the left and right hand sides as 2 to the something. So now that each side is written with the same base we know the exponents must be equal. Check:

9. Example 1: (Since the bases are the same we simply set the exponents equal.) Here is another example for you to try: Example 1a:

10. Example 2: (Let’s solve it now) (our bases are now the same so simply set the exponents equal) Let’s try another one of these.

11. Example 3 Remember a negative exponent is simply another way of writing a fraction The bases are now the same so set the exponents equal.

12. All of the transformations that you learned apply to all functions, so what would the graph of look like? up 3 up 1 Reflected over x axis down 1right 2

13. Reflected about y-axisThis 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. To model these we need to learn about a special base.

14. Slide 3- 14 The Nature of Exponential Functions A Table of Values

15. The Base “e” (also called the natural base) To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e 1. You do this by using the e x button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the e x, you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the e x. You should get 2.718281828 Example for TI-83

17. To Do Complete pg 247- 38, 40, 42, 44 Analyze Domain, Range, Continuity, Decreasing, Increasing, Symmetry(even, odd), Bounded, Extrema, Horizontal Asymptotes, Vertical Asymptotes, Using limits describe behavior of the function as x approaches the vertical asymptote, End behavior Pg 286 #2, 4, 6, 12, 24, 66 Homework: pg 287 1-19 odds Read Sec. 3.2

18. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.www.mathxtc.com Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au