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:
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)
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)
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)
and remove the inner parentheses:
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)
The vertical asymptote is still x = -4.
What is the equation of the vertical asymptote of the transformed function?
Answer: x = -4
Edwin
.
The parent function f(x)=log10x is
vertically stretched by a factor of 3,
reflected in the y-axis,
horizontally shifted 4 units to the left
and vertically 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) = .
Solved.
-------------
The post-solution note
This problem' solution assumes that you read the problem attentively and solve it adequately.