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MCR3U1 U6L5 TRANSFORMATIONS OF EXPONENTIAL FUNCTIONS (using the Transformation Rules & Mapping) PART A ~ INTRODUCTION Any function, 𝑦 = 𝑓(𝑥), can be transformed according to: 𝑦 = 𝑎𝑓[𝑘(𝑥 − 𝑑)] + 𝑐 Transformations are applied to an exponential function, 𝑦 = 𝑏 𝑥 , as follows: 𝑦 = 𝑎𝑏𝑘(𝑥−𝑑) + 𝑐 where 𝑎 determines vertical stretches/compressions/reflections 𝑘 determines horizontal stretches/compressions/reflections 𝑑 determines horizontal translations (right/left) 𝑐 determines vertical translations (up/down) (NOTE: The horizontal asymptote changes when vertical transformations are applied.) A mapping formula may also be used: 𝑥 (𝑥, 𝑦) → ( + 𝑑, 𝑎𝑦 + 𝑐) 𝑘 PART B ~ ANATOMY OF THE TRANSFORMED EXPONENTIAL FUNCTION If 𝑓(𝑥) = 2𝑥 , determine an equation for 𝑔(𝑥) if 𝑔(𝑥) = −4𝑓(−3𝑥 − 9) + 6. Write a mapping formula and describe each of the transformations that have occurred. 𝑔(𝑥) = (𝑥, 𝑦) → Inquiring Minds Want to Know!! Consider 𝑓(𝑥) = 23𝑥 and 𝑔(𝑥) = 8𝑥 . How are these functions related???? MCR3U1 U6L5 PART C ~ GRAPHING EXAMPLES Graph each of the following functions. Include a sketch of the parent function, state the y–intercept, the equation of the asymptote, the domain, and the range. a) 𝑦 = −2𝑥 + 4 y x b) y–intercept: ______________________________ asymptote: ______________________________ domain: __________________________________ range: ____________________________________ 𝑦 = 3−(𝑥−4) y x y–intercept: ______________________________ asymptote: ______________________________ domain: __________________________________ range: ____________________________________ MCR3U1 c) U6L5 1 3𝑥 𝑦=( ) 2 y −2 (x, y) x y–intercept: ______________________________ asymptote: ______________________________ domain: __________________________________ range: ____________________________________ HOMEWORK: p.251–253 #1, 4, 5ac, 9, 10 Graph each of the following exponential functions and determine: i) the equation of the horizontal asymptote ii) the y–intercept iii) the domain and range a) 𝑦 = −2(3𝑥 ) − 1 b) 1 −𝑥−2 𝑦=( ) 3