Question 1181202
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The parent function f(x)=log<sub>10</sub>x is 
vertically stretched by a factor of 3, 
reflected in the y-axis, 
horizontally {{{highlight(cross(transformed))}}} <U>shifted</U> 4 units to the left
and vertically {{{highlight(cross(transformed))}}} <U>shifted</U> 2.5 units up. 

What is the equation of the vertical asymptote of the transformed function?
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;First, &nbsp;look how I edited your post to make words/terms usage &nbsp;CONSISTED &nbsp;and mathematically correct.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Second, &nbsp;when we solve such problem, &nbsp;there is &nbsp;NO &nbsp;NEED &nbsp;to trace the concrete 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;function  &nbsp;f(x)=log<sub>10</sub>x  &nbsp;behavior/transformations/changes,


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We should trace/analyse/follow the behavior of the vertical asymptote, &nbsp;ONLY.



<pre>
So, I look for changes of the vertical asymptote, ONLY, at each given step.


(a)  Vertical stretch by a factor of 3 DOES NOT CHANGE vertical asymptote x= 0.

     After this step, the asymptote REMAINS to be vertical x= 0.



(b)  Reflection in the y-axis DOES NOT CHANGE vertical asymptote x = 0.

     After this step, the asymptote REMAINS to be vertical x= 0.



(c)  Horizontal shift 4 units to the left  DOES CHANGE vertical asymptote x = 0.

     After this step, the asymptote BECOMES to be vertical x= -4.



(d)  Vertical shift 2.5 units up DOES NOT CHANGE vertical asymptote x= - 4.

     After this step, the asymptote REMAINS to be vertical x= -4.



<U>ANSWER</U>.  After all the listed transformations, the original vertical asymptote x= 0 becomes vertical asymptote x= -4, finally.


         It is true not only for the given parent function f(x).  

         With the given steps, it is true for ANY OTHER parent function with the vertical asymptote x= 0: 

             for example, for the parent function y = f(x) = {{{1/x}}}.
</pre>

Solved.


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<H3>The post-solution note</H3>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This problem' solution assumes that you read the problem attentively and solve it adequately.