.2 Transformations of Exponential Functions. Math

Transcription

1 .2 Transformations of Eponential Functions Math

2 Vertical Translation Given the graph of f() = 2 g() = Shifts the graph up if k > 0. The graph of f() moves upward 3 units. (, y) (, y + k) (0, 1) (0, 4) f() = c + k g ( ) 2 3 f( ) 2 Horizontal Asymptote h() = 2 4 Shifts the graph down if k < 0. The graph of f() moves downward 4 units. (, y) (, y + k) (0, 1) (0, 3) h ( ) 2 4 Math

3 Horizontal Translation Given the graph of f() = 2 g() = Shifts the graph to the left if h < 0. The graph of f() moves to the left 3 units. (, y) ( + h, y) (0, 1) ( 3, 1) h() = 2 4 Shifts the graph to the right if h > 0. The graph of f() moves to the right 4 units. (, y) ( + h, y) (0, 1) (4, 1) f() = c h g ( ) 2 3 f( ) 2 h ( ) 2 4 Math

4 7.2 Eponential Functions Move to page 3.1. Sketch a possible graph for each of the following. Label the y-intercept Graph: f() = ab a > 0 a < 0 When b > 1 When b = 1 When 0 < b < 1 Math

5 Vertical Stretch Given the graph of f() = 2 f() = ac f( ) 2 g() = 4(2) g ( ) 4(2) Vertical stretch about the -ais by a factor of 4. (, y) (, ay) (0, 1) (0, 4) g() = 4(2) For a < 0, there is a reflection in the -ais. (, y) (, ay) (0, 1) (0, 4) h ( ) 4(2) Math

6 Horizontal Stretch f() = c b Given the graph of f() = 2 g() = 2 4 Horizontal stretch about the y-ais by a factor of 1. 1 (, y) (, y) 4 b 1 2 (2, 4) (, 4) h ( ) 2 4 g ( ) 2 4 f( ) 2 g() = 2 4 For b < 0, there is a reflection in the y-ais. (, y) (, ay) 1 b 1 2 (2, 4) (, 4) Math

7 Transformations Involving Eponential Functions McGraw Hill DVD Resources N05_7.2_348_IA Transformation Equation Description Horizontal stretch g() = c b Horizontal stretch about the y-ais by a factor of. 1 b Vertical stretch Reflecting g() = a c g() = c Vertical stretch about the -ais by a factor of a. Multiplying y-coordintates of f () = c by a. Reflects the graph of f () = c about the -ais. g() = c - Reflects the graph of f () = c about the y-ais. Vertical translation Horizontal translation g() = c + k g() = c -h Shifts the graph of f () = c upward k units if k > 0. Shifts the graph of f () = c downward k units if k < 0. Shifts the graph of f () = c to the right h units if h > 0. Shifts the graph of f () = c to the left h units if h < 0. Math

8 Apply Transformations to Sketch a Graph f() = a(c) b( h) + k Consider the eponential function equation What is the base function related to g()? g 4 ( ) 2(3) 4 f( ) (3) Describe a sequence of transformations required to transform the graph of the base function to the graph of g(). Write the transformation in mapping notation for the point (, y). Vertically stretched by a factor of 2 Horizontally stretched by a factor of ¼ Vertical translation 4 units down. (, y),2( y 4) 4 0,1 0,6 Horizontal Asymptote y 4 On the net page, complete the table to list the coordinates of the image points after the transformation. Describe the effects on the domain, range, equation of the horizontal asymptote, and intercepts after the transformation. Math

9 Apply Transformations to Sketch a Graph f() = a(c) b( h) + k g 4 ( ) 2(3) 4 f( ) 3 f( ) 3 (0, 1) (1, 3) (2, 9) (3, 27) (4, 81) g 4 ( ) 2(3) 4 (0, 6) 1,10 4 1,22 2 3,58 4 1,166 Domain remains the same: { R} Range becomes:{ y y 4, y R} Equation of the horizontal asymptote is y = 4. No -intercepts. The y-intercept Math 30-1 is 6. 9

10 Which of the following transformations of the graph of would result in the y-intercept being invariant? y 2 1 1,1 y 2 3 0,1 y 2 0,1 y 3 2 0,3 y ,9 Determine the value of the missing coordinate. The point (a, 9) is on the graph of y 2 1 The point (a, 27) is on the graph of y 3 1 a = 3 a = 4 Math

11 Writing an Eponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. Math

12 Writing an Eponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. What is the population after 5 years? P = a(b) t P = 20(3) t = a is the initial amount b is the growth factor t is time in years Eponential growth model Substitute a, b Substitute t = 5. Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years. Math

13 Some household smoke detectors contain a small amount of the radioactive element Americium 241. They are designed to detect hot, fast-burning fires (think grease fire). Canadian Nuclear Safety and Control Act on Nuclear Substances and Radiation Devices Regulations does not consider smoke detectors to be radioactive waste based on the minimal amount of radiation they put out, The life span of a smoke detector is approimately 10 years. Where should you recycle the used smoke detectors? Am-241 has a half-life of approimately 432 years. The average smoke detector contains 200µg of Am-241. Write an eponential function that models the decay of Am-241. y t p a b 432 y t Math

14 Tetbook p Level 1: (Basic Drill and Practice) 1 6 Level 2: (Problem Solving) 7, 8, 9, 10, Level 3: (Etension and Higher Level) 11, 12, 13, 14, Math