Question 1190912
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For clarity, rewrite the new function as<br>
{{{y=-f((1/4)x+6)-7}}}<br>
Then rewrite the new function in the form<br>
{{{y=a*f(b(x-c))+d}}}<br>
The transformations will be, in order,<br>
(1) translation of c units
(2) horizontal compression by a factor of b
(3) vertical stretch by a factor of a
(4) vertical translation by d<br>
Note the "(1/4)x+6" needs to be rewritten as "(1/4)(x+24)".<br>
The rest is straightforward....<br>
{{{y=-f((1/4)x+6)-7}}} --> {{{y=(-1)(f((1/4)(x-(-24))))+(-7)}}}<br>
That gives us a=-1, b=1/4, c=-24, and d=-7<br>
The transformations are then, in order,
(1) translation 24 units left
(2) horizontal compression by a factor of 1/4
(3) vertical stretch by a factor of -1 (i.e., reflect over the x-axis)
(4) vertical translation down 7<br>
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In reply to the student's comment....<br>
A horizontal compression by a factor of a is a horizontal stretch by a factor of 1/a.  So in this example, the compression by a factor of 1/4 is a stretch by a factor of 4.<br>
That is the correct interpretation.<br>
As an example, consider f(x)=sin((1/4)x).  The compression factor is 1/4 which means there is a horizontal stretch by a factor of 4.<br>
This is correct, because the function sin((1/4)x) completes a cycle every 8pi ({{{0<=(1/4)x<-2pi}}} --> {{{0<=x<=8pi}}}), while the function sin(x) completes a cycle every 2pi -- the graph is stretched horizontally by a factor of 4.<br>