Question 147685
If g(x) is a logarithmic function, then what effect does 'c' have on g(x) if f(x) = g(cx)?
a. g(x) shifts c units to the sign opposite of c
b. g(x) stretches or shrinks horizontally
c. g(x) stretches vertically
d. There is no effect on g(x)
<pre><font size = 4 color = "indigo"><b>

This is a poorly written problem.  If 'c' were = 1 then there
is no effect and the answer would be d.  If 'c' were negative 
there would also be a reflection across the y-axis.  But none
of the answers allows for this. 

Here is the graph of  

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

{{{graph(400,400,-5,5,-5,5,log(3,x))}}}

Here is the graph of  

{{{f(x) = g(2x)=log(3,(2x)))}}}

{{{graph(400,400,-5,5,-5,5,log(3,2x))}}}

You can see that the the second graph has been shrunk horizontally.

--------------------------------------------

Here is the graph of  

{{{g(x) = log(2,(x))}}}

{{{graph(400,400,-5,5,-5,5,log(3,x))}}}

Here is the graph of  

{{{f(x) = g(0.5x)=log(2,(0.5x))}}}

{{{graph(400,400,-5,5,-5,5,log(3,0.5x))}}}

You can see that the the second graph has been stretched horizontally.

-------------------------------------------------

Here is the graph of  

{{{g(x) = log(4,(x))}}}

{{{graph(400,400,-5,5,-5,5,log(4,x))}}}

Here is the graph of  

{{{f(x) = g(-2x)=log(4,(-2x))}}}

{{{graph(400,400,-5,5,-5,5,log(4,-2x))}}}

You can see that the the second graph has been shrunk horizontally
and reflected across the y-axis.

----------------------------------------

Here is the graph of  

{{{g(x) = log(5,(x))}}}

{{{graph(400,400,-5,5,-5,5,log(5,x))}}}

Here is the graph of  

{{{f(x) = g(0.5x)=log(5,(-0.5x))}}}

{{{graph(400,400,-5,5,-5,5,log(5,-0.5x))}}}

You can see that the the second graph has been stretched horizontally
and reflected across the y-axis.

Therefore, the correct choice is probably (b), but that does not tell 
about the fact that if c is negative there is also a refection across the
y-axis

There is no effect when {{{c=1}}}
There is a horizontal shrink when {{{c>1}}}
There is a horizontal stretch when {{{0<c<1}}}
There is both a horizontal shrink and a refection across the y-axis when 
{{{c<-1}}}
There is a horizontal stretch and a reflection across the y-axis when {{{-1<c<0}}}
There is only a reflection across the y-axis when {{{c=-1}}}

More of these possibilities involve horizontal shrinks and stretches, so 
your teacher probably intended for the correct choice to be b.

Edwin</pre>