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**Lesson Focus:**

To apply translations, stretches and reflections to the graphs of exponential functions; to represent these transformations in the equations of exponential functions.

you can use transformed exponential functions to model real-world applications of exponential growth or decay to graph an exponential function of the form

*y*

*k*

, apply transformations to the base function

*y*

*c x*

, where

*c*

0

each of the parameters

*a*

,

*b*

,

*h*

and

*k*

is associated with a particular transformation

vertical stretch about the

*x*

-axis by a factor of

*a a *

*a*

0 results in a reflection in the

*x*

-axis

*y*

*x b h k *

horizontal stretch about the

*y*

-axis by a factor of

1

*b*

*b*

0 results in a reflection in the

*y*

-axis

*x*

,

*b y*

horizontal translation

vertical translation

*k*

*y y*

*y*

2

*x x*

2

*x*

2

when applying transformations by graphing, you must apply the parameters

*a*

and

*b*

before the parameters

*h*

and

*k*

(remember the acronym

) if performing transformations using point transformations, remember that

*x b*

*k*

For each of the transformed functions: a) b) c) d) state the parameter and describe the transformation graph the base function and the transformed function on the same grid describe any changes to the domain, range, intercepts and equation of the horizontal asymptote give the point transformation i)

*y*

4 2

3

*x*

ii)

*y*

*x*

2

Describe the transformations of

*y*

4

*x*

caused by

*y*

4

1

3

*x*

3

1 .

The real estate board in a city announces that the current average price of a house in the city is

$400 000. It predicts the average prices will double every 15 years.

a) Write a transformed exponential function in the form

*y*

*k*

to model this situation. b) c)

Describe how each of the parameters in the transformed function relates to the information provided.

Draw a sketch of the graph. d) Predict the value of a house after 10 years. e) How long will it take for the average house to

have a value of $500 000?