SOLUTION: The parent function f(x)=log<sub>10</sub>x is vertically stretched by a factor of 3, reflected in the y-axis, horizontally transformed 4 units to the left and vertically transfor

Algebra ->  Trigonometry-basics -> SOLUTION: The parent function f(x)=log<sub>10</sub>x is vertically stretched by a factor of 3, reflected in the y-axis, horizontally transformed 4 units to the left and vertically transfor      Log On


   

Question 1181202: The parent function f(x)=log10x is vertically stretched by a factor
of 3, reflected in the y-axis, horizontally transformed 4 units to the left
and vertically transformed 2.5 units up. What is the equation of the
vertical asymptote of the transformed function?
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The parent function f(x)=log10x is vertically stretched by a factor
of 3, reflected in the y-axis, horizontally transformed 4 units to the left
and vertically transformed 2.5 units up. What is the equation of the
vertical asymptote of the transformed function?
The parent function:
f%28x%29=log%2810%2C%28x%29%29 

The vertical asymptote is the y-axis, whose equation is x = 0.

is vertically stretched by a factor of 3,

So we multiply the entire right side by 3, and call it g(x)

g%28x%29=3log%2810%2C%28x%29%29 

 

The vertical asymptote is still the y-axis, whose equation is x = 0.

reflected in the y-axis,
 
We replace x by -x, and label it h(x)

h%28x%29=3log%2810%2C%28-x%29%29 

 

The vertical asymptote is still the y-axis, whose equation is x = 0.

horizontally transformed 4 units to the left

We replace x by (x+4) and label the new function k(x)

k%28x%29=3log%2810%2C%28-%28x%2B4%29%5E%22%22%29%29

and remove the inner parentheses:

k%28x%29=3log%2810%2C%28-x-4%5E%22%22%29%29



Now the vertical asymptote has moved 4 units left, so its equation 
is x = -4.

and vertically transformed 2.5 units up.
 
We add 2.5 to the entire right side and label the new function m(x)

m%28x%29=3log%2810%2C%28-x-4%5E%22%22%29%29%2B2.5



The vertical asymptote is still x = -4.

What is the equation of the vertical asymptote of the transformed function?

Answer:  x = -4

Edwin

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
The parent function f(x)=log10x is
vertically stretched by a factor of 3,
reflected in the y-axis,
horizontally highlight%28cross%28transformed%29%29 shifted 4 units to the left
and vertically highlight%28cross%28transformed%29%29 shifted 2.5 units up.
What is the equation of the vertical asymptote of the transformed function?
~~~~~~~~~~~~~~~~~~~


            First,  look how I edited your post to make words/terms usage  CONSISTED  and mathematically correct.

            Second,  when we solve such problem,  there is  NO  NEED  to trace the concrete
            function  f(x)=log10x  behavior/transformations/changes,

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


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.



ANSWER.  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%2Fx.

Solved.

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The post-solution note


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