Transformations

Question

Use a sequence of transformations, starting from the reciprocal function y=1/x to graph the rational function r(x) = (4x+3)/(2x+1) Show each graph in the sequence and describe each transformation.

Hint: use long division to write r(x) as a quotient plus proper rational function, r(x) =A+B/2x+1

Answer 1

Using long division:

To do the long division, start by asking how many times does 2x go into 4x? The answer is, of course, 2 times. Put the 2 on top and multiply the 2 by the "2x" and then by the "1" and put it below the "4x + 3" then subtract. You get "1" so "1" is the remainder. The answer then is 2 remainder 1. But you can also write it as 2 + 1/(2x + 1)

So r(x) = 1/(2x + 1) + 2

To understand the transformation though you must put it in "standard transformation form" by factoring out the "2" in front of the "x":

r(x) = 1/[2(x + 1/2)] + 2

When doing the transformation remember that what happens to the "x" is opposite the way it appears. So adding moves the graph left, multiplying by a constant "a" is a compression by "1/a"

The transformation then from y = 1/x is:

1. Horizontal compression by a factor of 1/2
2. Horizontal translation 1/2 unit to the left
3. Vertical translation 2 units up

Meaning, the point (2, 0.5) that is on y = 1/x transforms the x-value (2) by multiplying it by 1/2 then moving it left by 1/2 resulting in the "new x" being 1/2. The y-value (0.5) moves up 2 making it 2.5 so the point (2, 0.5) transforms to (0.5, 2.5) on r(x).