Question 1181202
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 transformed 2.5 units up. What is the equation of the
vertical asymptote of the transformed function?

The parent function: 

{{{f(x)=log(10,(x))}}}

 {{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10,ln(x)/ln(10)) )}}}

<pre>The vertical asymptote is the y-axis, whose equation is x = 0.</pre>
is vertically stretched by a factor of 3,<pre>

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

{{{g(x)=3log(10,(x))}}} 

 {{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10,ln(x)/ln(10),3ln(x)/ln(10)) )}}}

The vertical asymptote is still the y-axis, whose equation is x = 0.</pre>
reflected in the y-axis,<pre> 
We replace x by -x, and label it h(x)

{{{h(x)=3log(10,(-x))}}} 

 {{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10,ln(x)/ln(10),3ln(x)/ln(10),
3ln(-x)/ln(10)
) )}}}

The vertical asymptote is still the y-axis, whose equation is x = 0.</pre>
horizontally transformed 4 units to the left<pre>

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

{{{k(x)=3log(10,(-(x+4)^""))}}}

and remove the inner parentheses:

{{{k(x)=3log(10,(-x-4^""))}}}

{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10,ln(x)/ln(10),3ln(x)/ln(10),
3ln(-x)/ln(10),3ln(-x-4)/ln(10)
) )}}}

Now the vertical asymptote has moved 4 units left, so its equation 
is x = -4.</pre>
and vertically transformed 2.5 units up.<pre> 
We add 2.5 to the entire right side and label the new function m(x)

{{{m(x)=3log(10,(-x-4^""))+2.5}}}

{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10,ln(x)/ln(10),3ln(x)/ln(10),
3ln(-x)/ln(10),3ln(-x-4)/ln(10), 3ln(-x)/ln(10),11,3ln(-x-4)/ln(10)+2.5
) )}}}

The vertical asymptote is still x = -4.</pre>
What is the equation of the vertical asymptote of the transformed function?<pre>

Answer:  x = -4

Edwin</pre>