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Exponential Function
1.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Copyright © 2012 JOVIC G. RULLEPA All Rights Reserved
2.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Copyright © 2012 JOVIC G. RULLEPA All Rights Reserved.
3.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016
4.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Objectives: 1. Define an exponential function; and 2. Identify an exponential function through its (a) table of values, (b) graph and (c) equation.
5.
K-12 SHS Copyright
© by J. G. Rullepa. All rights reserved.2016
6.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Moral of the Story: “Understanding the value of small things as a foundation of doing great things.”
7.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Group Activity
8.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Situation 1: Shown at the left is a pattern of “growing” squares from match sticks. 1.Study the pattern and draw a picture of the next likely figure in the pattern. 2.Make a table of values showing the number of matchsticks on one side of the figure and the perimeter of the figure. 3.Sketch the graph of the relationship shown in the table (side length, perimeter) 4.Describe in words the relationship between the perimeter and the side of each figure formed. No. of matchsticks on one side of the figure 1 2 3 4 5 6 Perimeter of the figure 4 8 12
9.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Situation 2: Shown at the left is a pattern of “growing” squares from match sticks. 1. Study the pattern and draw a picture of the next likely shape in the pattern. 2. Make a table of values showing the number of matchsticks on one side of the figure and the area of the figure in square units. 3. Sketch the graph of the relationship shown in the table (side length, area) 4. Describe in words the relationship between the area and the side of each figure formed. No. of matchsticks on one side of the figure 1 2 3 4 5 6 Area of the figure in square units 1 4 9
10.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Situation 3: Shown at the left is a pattern of “growing” squares from match sticks where the length of the side was doubled every time. 1. Study the pattern and draw a picture of the next likely shape in the pattern. 2. Make a table of values showing the area of each square figure in square units. 3. Sketch the graph of the relationship shown in the table (order of square figure, area) 4. Describe in words the relationship between the area and the order of square figures. Order of square figures 1 2 3 4 5 6 Area of the figure in square units 1 4 16
11.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Discussion Solution for #1 Solution for #2 Solution for #3
12.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 On the story of a chessboard, the pattern of increasing the number of wheat grains on each square of a chess board is presented by the table below: Noticed that the number of grains for each squares can be broken down as a power of 2. Number of squares 1 2 3 4 5 6 7 8… Number of grains 1 2 4 8 16 32 64 128 …
13.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Order of squares No. of grains Breakdown of y as a power of 2 1 1 = 20 2 2 2 =21 3 4 2 2 =22 4 8 2 2 2 =23 5 16 2 2 2 2 =24 6 32 2 2 2 2 2 =25 7 64 2 2 2 2 2 2 =26 8 128 2 2 2 2 2 2 2 =27 : : 64 9.22337 x 1018 2 2 2 2 2 2 … 2 =263 : : x ∞ 2 2 2 2 … 2 = 2x
14.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 The graph of f(x) = 2x x - values -3 -2 -1 0 1 2 3 4 f(x) 1/8 1/4 1/2 1 2 4 8 16
15.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Definition: • An exponential function can be written as • Where b > 0, b 1, and x is any real numbers power base exponent
16.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 The graph of f(x) = 2x x-values -3 -2 -1 0 1 2 3 4 f(x) 1/8 1/4 1/2 1 2 4 8 16 The graph of f(x) = (1/2)x x-values -3 -2 -1 0 1 2 3 4 f(x) 8 4 2 1 1/2 1/4 1/8 1/16
17.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Important Notes: We identify exponential function by a. Table of Values – if there is no equal differences in terms of y. b. Graph – if b > 1, the graph is increasing rapidly to the right and coming closer and closer to the x-axis to the left. - if 0 < b < 1, the graph is increasing rapidly to the left and coming closer and closer to the x-axis to the right. c. Equation – the variable is on the exponent.
18.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Check This Out: Consider a game called Tower of Hanoi. It consists of circular discs of graduated sizes inserted in one of the three pegs. The problem is to transfer the discs to any of the other pegs in the fewest number of possible moves with no bigger disk placed above a smaller disk and a disks can only be moved one at a time. a. Complete the table below. b. Sketch the graph of the relationship shown on the table (x,y). c. Find the relation between x and y.Number of Disks (x) 1 2 3 4 5 6 7 No. of Moves (y) 1 3 63 127
19.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Generalization
20.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Application: 1. Determine whether the following table of values describes an exponential function or not. a. b. c. d. x -2 -1 0 1 2 3 f(x) -4 -2 0 2 4 6 x 0 1 2 3 4 5 f(x) 1 3 9 27 81 243 x -2 -1 0 1 2 3 f(x) 12 3 0 3 12 27 x 0 1 2 3 4 5 f(x) 1 4 16 64 256 1024 Not exponential Not exponential Exponential Exponential
21.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Application: 2. Determine whether the following graph is an exponential function or not. a. b. c. d. Not exponential Not exponential Exponential Exponential
22.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Application: 3. Determine whether the following function is an exponential function or not. a. f(x) = -3x b. f(x) = (-3)x - 3 c. f(x) = x2 + 1 d. f(x) = 2/x e. f(x) = 4x+3 f. f(x) = 4xx g. f(x) = (x-2)-1 h. f(x) = 8(-x) Exponential Not exponential Not exponentialExponential Not exponential Not exponential Not exponential Exponential
23.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Evaluation: 1. Tell whether each statement is True or False. a. f(x) = 3x2 is an exponential function. b. If f(x) = 2x, then f(-1) = 2. c. If f(x) = (1/2)x, then f(-1) = 2. d. The function f(x) = 1x is an exponential function. e. The function f(x) = (-3)x is an exponential function.
24.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Evaluation: 2. Determine whether the given function represents an exponential function or not. a. y – 7 = x3 + 3 b. 5y = 43x c. 2x–4 = y + 3 d. 3 + 5x+2 = 4y
25.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016 Evaluation: 3. Given the following tables of ordered pairs. Identify those which exhibit an exponential trend. a. b. c. d. e. f. x 1 2 3 4 y 5 25 125 625 x -3 -2 -1 0 y 6 4 2 0 x -1 0 1 2 y -1 0 1 8 x 1 2 3 4 y 6 18 54 162 x 3 4 5 6 y 1/8 1/16 1/32 1/64 x 4 3 2 1 y 16/3 8/3 4/3 2/3
26.
K-12 SHS Math
Copyright © by J. G. Rullepa. All rights reserved.2016
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