Exploration of Exponential Functions
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1 Eploration of Eponential Functions
2
3 Prior Knowledge If a is any positive number and is any integer, then a 0 If a is any positive number and is any integ 4 e.g e.g Understand and apply the laws of indices Functions
4 Eponents/Indices/Powers (base) eponent Base 5 Eponent / inde / power The eponent says how many of the base are being multiplied together.
5 John has 0 to buy bars of chocolate which cost each. Bars bought Amount of Money left
6 John has 0 to buy bars of chocolate which cost each. Let be the number of chocolate bars we buy and y be the amount of money left. The relationship between and y is shown below y Each value of corresponds to values of y. Each value of corresponds to only one value of y & y is a function of if both of these conditions are true
7 .Each value of corresponds to values of y 4 4 &.Each value of corresponds to only one value of y 4 4 4
8 .Each value of corresponds to values of y 4 4 & Fails condition.each value of corresponds to only one value of y 4 Fails condition 4 4 Fails both conditions
9 .Each value of corresponds to values of y 4 Bijective &.Each value of corresponds to only one value of y
10 Basic Technique Read information from a graph e.g. the figure shows the graph of p() = + and q() = - in the domain -4 4, R Discuss what are the similarities and the differences
11 Basic Technique Read information from a graph e.g. the figure shows the graph of P() = + in the domain -4 4, R The graph has no - intercepts Its y intercept is For 0, + increases as increases For 0, + increases as decreases
12 Eponential Functions Learning Outcomes After completing this session you will be able to:. Understand the properties of eponential functions. Learn the features of their graphs
13 Eponential Functions Activity Sheets Section A SA:-4 Section B SA:-4 Section C SA: -4 Page 4-8 Page 9 - Page 4-6 y = & y= Compare Characteristics y=(/) & y=(/) Compare Characteristics Compare y=(/) and y= Compare y=(/) and y= Now I see. Section 4 Page 7-8 Problem Solving Questions
14 Organisation Groups & 5 Complete Section A Student Activity page 4-5 Groups, 7 & 9 Complete Section B Student Activity page 9-0 Groups & 6 Complete Section A Student Activity page 6-7 Groups 4,8 & 0 Complete Section B Student Activity page -
15 Feedback Groups & 5 Section A Student Activity page 4-5 Groups & 6 Section A Student Activity page 6-7
16 Feedback Q (i) Base f = g = Base (ii) Eponent (iii) Varying (iv) Constant Eponent Varying Constant Q Domain Domain 4,5,6 & 7
17 Feedback Q (i) Base f = g = Base (ii) Eponent Eponent (iii) Varying & f() Varying & g() (iv) Constant Constant Q Domain R Domain R 4,5,6 & 7
18 () X f() () X g() -4 () -4 /6 - () - /8 - () - /4 - () - / 0 () 0 () () () 8 4 () () -4 /8 - () - /7 - () - /9 - () - / 0 () 0 () () 9 () 7 4 () 4 8 4,5,6 & 7
19 Groups & 5 & 6 f = g = 4,5,6 & 7
20 Groups & 5 & 6 f = g = 4,5,6 & 7
21 Groups, 5, & 6 g = f = 4,5,6 & 7
22 Questions 4 & 5 Q4 Q5 (i) Straight Line No (i) Outputs (Range) R + (ii) Y increasing or decreasing as Increasing (ii) Negative Outputs, Why? No increases (iii) Rate of change Not constant (iii) Outputs as decreases Decreases (iv) Describe how its curvature/rate of Increasing (iv) An output of 0? Why? No change is changing (v) X-intercept None (vi) Y-intercept 4,5,6 & 7
23 Questions 4 & 5 Q4 Q5 (i) Straight Line (i) Outputs (Range) (ii) Y increasing or decreasing as increases (iii) Rate of change (iv) Describe how its curvature/rate of change is changing (ii) Negative Outputs, Why? (iii) Outputs as decreases (iv) An output of 0? Why? (v) X-intercept (vi) Y-intercept 4,5,6 & 7
24 Feedback Groups, 7 & 9 Section B Student Activity page 9-0 Groups 4,8 & 0 Section B Student Activity page -
25 Feedback Q (i) Base f = Base g = (ii) Eponent Eponent (iii) Varying & f() Varying & g() (iv) Constant Constant Q Domain R Domain R 9,0, &
26 f() g() ,0, &
27 Groups,7 & 9 4,8 & 0 f = g = 9,0, &
28 Groups,7 & 9 4,8 & 0 g = f = 9,0, &
29 Groups,7,9,4,8 & 0 g = f =
30 Questions 4 & 5 Q4 Q5 (i) Straight Line (i) Outputs (Range) (ii) Y increasing or decreasing as increases (iii) Rate of change (iv) Describe how its curvature/rate of change is changing (ii) Negative Outputs, Why? (iii) Outputs as decreases (iv) An output of 0? Why? (v) X-intercept (vi) Y-intercept 9,0, &
31 Questions 4 & 5 Q4 Q5 (i) Straight Line No (i) Outputs (Range) R + (ii) Y increasing or decreasing as Decreasing (ii) Negative Outputs, Why? No increases (iii) Rate of change Not constant (iii) Outputs as decreases Increases (iv) Describe how its curvature/rate of Decreasing (iv) An output of 0? Why? No change is changing (v) X-intercept None (vi) Y-intercept 9,0, &
32 Organisation Groups, 5, & 6 Draw either or Complete Section A Activities & 4 Page 8 Groups,7,9,4,8,& 0 Draw either or Complete Section B Activities & 4 Pages & 4
33 Compare and Section A Activity page 8 (Groups,,5,& 6). How are they the same and how are they different?. Are they functions?. Name this type of function and why? 8
34 Understand the characteristics of f( ) a, a Section A Activity 4 page 8 Domain Curvature Straight Line Is y increasing or decreasing as increases Range X-intercept Y-intercept Maimum/ Minimum value 8
35 Understand the characteristics of f( ) a, a Section A Activity 4 page 8 Domain R Curvature Increasing Straight Line No Range R + Is y increasing or X-intercept None decreasing as increases Increasing Y-intercept Maimum/ None Minimum value 8
36 Compare and Section B Activities & 4 page. How are they the same and how are they different?. Are they functions?. Name this type of function and why?
37 Understand the characteristics of f( ) a,0 a Section B Activity 4 page Domain Curvature Straight Line Is y increasing or decreasing as increases Range X-intercept Y-intercept Maimum/ Minimum value
38 Understand the characteristics of f( ) a,0 a Section B Activity 4 page Domain R Curvature Decreasing Straight Line No Range R + Is y increasing or X-intercept None decreasing as increases Decreasing Y-intercept Maimum/ None Minimum value
39 Organisation Groups,,5,7,& 9 Section C Activity Page 4 Groups,4,6,8 & 0 Section C Activity Page 4
40 Compare and. Same. Different. Write f = using a base of 4. What transformation maps the graph of f = onto the graph of f =? 4
41 Compare and. Same. Different. Write g = using a base of 4. What transformation maps the graph of g = onto the graph of g =? 4
42 Compare f = f = g = g =
43 All Groups Complete Section C Activity page 5
44 Section C. If f = a, a R, a >, then the properties of the eponential function are:. If f = a, a R, a >, then the features of the eponential graph are:. If f = a, a R, 0 < a <, then the properties of the eponential function are: 4. If f = a, a R, 0 < a <, then the features of the eponential graph are: 5
45 All Groups 6
46
47 f( ) a,0 a Eponential Functions (properties) ( ) f a, a Eponential Graphs (features) and and
48 Prior Knowledge Connections Effective questioning What if questions Underlying Principles Group work Misconceptions Discussion & Communication Methods rather than answers Rich tasks
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