Video Transcript
Use the shown graph of 𝑦 equals 𝑓
of 𝑥 to find 𝑎, 𝑏, and 𝑐 so that 𝑓 of 𝑥 equals 𝑎 times the absolute value of
𝑥 minus 𝑏 plus 𝑐.
By visual inspection, we can see
that 𝑓 of 𝑥 is some kind of transformed absolute value function. Sketching in 𝑔 of 𝑥, the standard
absolute value function, will help us to see what transformations have occurred. First of all, we can see that the
slope of the graph has changed. It has been stretched upwards or
squashed inwards, which amounts to the same thing.
Looking at the graphs, we can see
that the standard absolute value function has a slope of plus or minus one. This has changed to a slope of plus
or minus three for 𝑓 of 𝑥. Second, by looking at the vertices
of the graphs, we can see that 𝑓 has been shifted three units to the left. Finally, we can see that the graph
of 𝑓 of 𝑥 has shifted four units down.
The geometric transformation
changing the slope of the graph from one to three can be thought of as a vertical
dilation with a scale factor of three. Algebraically, this corresponds to
multiplying the absolute value function by three.
Notice that this transformation can
also be thought of as a horizontal dilation with a scale factor of one-third, that
is to say, squashing the graph horizontally rather than stretching it
vertically. Algebraically, this corresponds to
multiplying the variable 𝑥 by the reciprocal of one-third, that is, by three. Because the absolute value function
is a linear function passing through the origin, this amounts to the same
transformation.
The next transformation is a
translation in the 𝑥-direction by three units to the left. Algebraically, this corresponds to
adding three units to the variable 𝑥. The final transformation is a
vertical translation by four units down. Algebraically, this corresponds to
subtracting four from the whole function.
We have arrived at our transformed
function. 𝑓 of 𝑥 equals three times the
absolute value of 𝑥 plus three, or 𝑥 minus negative three, all minus four. Thus, 𝑎 equals three, 𝑏 equals
negative three, and 𝑐 equals negative four.
