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) →
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


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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