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1 3/7/3 OBJ: SWBAT graph rational functions and recognize eponential functions. Bell Ringer: Start notes for Eponential functions Homework Requests: pg 46 #-9 odds 37, 39, 4, 43 Homework: p86 #-9 odds Read Sect. 3. Announcements: Quiz net Week Worksheet for over the weekend. Turn UP! Maimize Academic Potential

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3 Eponential Functions An eponential function is a function of the form y a b, where a 0, b 0, and b, and the eponent must be a variable. constant a is the initial value of f() at = 0, b is the base pg 86 #, 4, 6

4 Let s eamine eponential functions. They are different than any of the other types of functions we ve studied because the independent variable is in the eponent / - /4-3 /8 f f BASE Recall what a negative eponent means: Let s look at the graph of this function by plotting some points

5 Pg 80 a> 0, b > eponential growth, 0<b< Eponential Decay

6 y All of the transformations that you learned apply to all functions, so what would the graph of y 3 look like? up 3 up y Reflected over ais y right down

7 Reflected about y-ais y This equation could be rewritten in a different form: y So if the base of our eponential function is between 0 and (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 eponential function. To model these we need to learn about a special base.

8 The Nature of Eponential Functions A Table of Values Determine formulas for the eponential function g and h whose values are given in the table below. Steps: f = a b a = f 0 b = f E: pg 87 # Because g is eponential, g( ) a b. Because g(0) 4, a 4. Because g() 4b, the base b 3. So, g( ) 4 3. Because h is eponential, h( ) a b. Because h(0) 8, a 8. Because h() 8 b, the base b / 4. So, h( ) 8. Slide 3-8 4

9 The Base e (also called the natural base) To model things in nature, we ll need a base that turns out to be between and 3. Your calculator knows this base. Ask your calculator to find e. You do this by using the e button (generally you ll need to hit the nd or yellow button first to get it depending on the calculator). After hitting the e, you then enter the eponent you want (in this case ) and push = or enter. If you have a scientific calculator that doesn t graph you may have to enter the before hitting the e. You should get Eample for TI-83

10 f e f 3 f

11 The Equality Property for Eponential Functions If a u = a v, then u = v This says that if we have eponential functions in equations and we can write both sides of the equation using the same base, we know the eponents are equal The left hand side is to the something. Can we re-write the right hand side as to the something? 34 3 Now we use the property above. The bases are both so the eponents must be equal We did not cancel the s, We just used the property and equated the eponents. You could solve this for now.

12 Let s try one more: We could however re-write both the left and right hand sides as to the something. 3 4 The left hand side is 4 8 to the something but the right hand side can t be written as 4 to the something (using integer eponents) So now that each side is written with the same base we know the eponents must be equal. Check:

13 Eample : (Since the bases are the same we simply set the eponents equal.) Here is another eample for you to try: Eample a: 3 3 5

14 Eample : (Let s solve it now) () (our bases are now the same so simply set the eponents equal) 3 3( ) Let s try another one of these.

15 Eample ( ) 5 Remember a negative eponent is simply another way of writing a fraction The bases are now the same so set the eponents equal. 4( )

16 To Do Complete pg 47-38, 40, 4, 44 Analyze Domain, Range, Continuity, Decreasing, Increasing, Symmetry(even, odd), Bounded, Etrema, Horizontal Asymptotes, Vertical Asymptotes, Using limits describe behavior of the function as approaches the vertical asymptote, End behavior Pg 86 #, 4, 6,, 4, 66 Homework: pg 87-9 odds Read Sec. 3.

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