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4.6 Transformations of Exponential Functions, Study notes of Mathematics

Saddleback CollegeMathematics

Graphing the Transformed Exponential Function. • The domain is always {x ε R}. Transformations do not change the domain. • The range depends on the location of ...

Typology: Study notes

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4.6 Transformations of Exponential Functions
All exponential functions can be written in the form:
f
(x) = a(b)k(x – d) + c,
where “b” is the base of the exponential equation, y = bx.
Example One
Determine the base function of following exponential functions and then determine the
transformations of each base function.
a)
f
(x) = 2(4)x + 5
b)
f
(x) = 0.2(1
2)3x – 10
Base Function:
Base Function:
Transformations:
Transformations:
pf3

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4.6 Transformations of Exponential Functions

All exponential functions can be written in the form:

f(x) = a(b)k(x^ –^ d)^ + c,

where “b” is the base of the exponential equation, y = bx.

Example One Determine the base function of following exponential functions and then determine the transformations of each base function.

a) f(x) = 2(4)x^ + 5 b) f(x) = 0.2(^12 )3x^ – 10

Base Function: Base Function:

Transformations: Transformations:

Note : on a graph of an exponential function, the asymptote will occur along the line y = c.

Example Two Determine the equations of the exponential function that have the following properties: a) Function One a. A base of 3 b. Vertically stretched by a factor of 2 c. Vertically translated 3 units up d. A reflection in the x-axis

b) Function Two a. A base of 2 b. Stretched vertically by a factor of 1. c. Reflected in the y-axis d. Its asymptote is the line y = 2

Graphing the Transformed Exponential Function  The domain is always {x ε R}. Transformations do not change the domain.  The range depends on the location of the horizontal asymptote and whether the function is above or below the asymptote. If it is above the asymptote, its range is y > c. If it is below, its range is y < c.  Apply each transformation one at a time, in the order: a, k, d, c.  You may need to factor the exponent to see what the exact transformations are. For

example, if the exponent is (2x + 2), then factor to get 2(x + 1) and then identifyk = 2

andd = -1.