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Chapter 8

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EXAMPLE 2 Connecting a geometric description of a<br />

function to an algebraic representation<br />

The logarithmic function y 5 log 10 x has been vertically compressed by a factor<br />

2<br />

of horizontally stretched by a factor of 4, and then reflected in the y-axis.<br />

3 ,<br />

It has also been horizontally translated so that the vertical asymptote is x 522<br />

and then vertically translated 3 units down. Write an equation of the transformed<br />

function, and state its domain and range.<br />

Solution<br />

y 5 a log 10 (k(x 2 d )) 1 c<br />

y 5 2 3 log 10a2 1 (x 1 2)b 2 3<br />

4<br />

Write the general form of the<br />

logarithmic equation.<br />

Since the function has been<br />

vertically compressed by a<br />

2<br />

factor of a 5 2 3 , 3 .<br />

Since the function has been<br />

horizontally stretched by a<br />

1<br />

factor of 4, so k 5 1 k 5 4, 4 .<br />

The function has been reflected<br />

in the y-axis, so k is negative.<br />

The vertical asymptote of the<br />

parent function is x 5 0.<br />

Since the asymptote of<br />

the transformed function is<br />

x 522, the parent function<br />

has been horizontally<br />

translated 2 units left,<br />

so d 522.<br />

The function has been vertically<br />

translated 3 units down, so<br />

c 523.<br />

Domain 5 5xPR 0 x ,226<br />

Range 5 5 yPR6<br />

The curve is to the left of the<br />

vertical asymptote, so the<br />

domain is x ,22.<br />

The range is the same as the<br />

range of the parent function.<br />

456 8.2 Transformations of Logarithmic Functions<br />

NEL

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