Just as a polynomial function $y=P\left(x\right)$y=P(x) can be transformed, so also can rational functions, given by $y=\frac{P\left(x\right)}{Q\left(x\right)}$y=P(x)Q(x), be transformed. Remember that transformation of a function just means to change it, we can transform a function using a translation (shift vertically or horizontally), dilation (expansion horizontally or vertically), or reflection (across an axis or other given line).
For example:
- The rational function $y=\frac{x-2}{x+5}$y=x−2x+5, expressible as $y=\frac{\left(x-2\right)}{\left(x-2\right)+7}$y=(x−2)(x−2)+7, can be thought of as the simpler function $y=\frac{x}{x+7}$y=xx+7 translated $2$2 units to the right.
- The rational function $y=\frac{2x^3}{x^2+5}+3$y=2x3x2+5+3 is similarly the function $y=\frac{2x^3}{x^2+5}$y=2x3x2+5 translated upwards by $3$3 units.
- The rational function $y=\frac{-12x}{x+1}$y=−12xx+1 is the function $y=\frac{x}{x+1}$y=xx+1 reflected in the $x$x axis and dilated by a factor of $12$12.
An Example
Find the equation of the transformed function created when the rational function $y=\frac{x^2}{x-1}$y=x2x−1 is first translated $3$3 units to the right and $5$5 units upward, and then dilated by a factor of $2$2.
Then the rational function $y=\frac{x^2}{x-1}$y=x2x−1 after translation becomes $y=\frac{\left(x-3\right)^2}{\left(x-3\right)+1}+5$y=(x−3)2(x−3)+1+5, and then after dilation becomes $y=\frac{2\left(x-3\right)^2}{\left(x-3\right)+1}+5$y=2(x−3)2(x−3)+1+5.
Note that the dilation does not influence the translation.
The transformed function can then be simplified to $y=\frac{2\left(x-3\right)^2}{x-2}+5$y=2(x−3)2x−2+5. The graph shows the two stages of transformation.
As an example, the point with coordinates $\left(-2,4\right)$(−2,4) on the original green curve translates to the point $\left(1,1\right)$(1,1) on the blue curve. Then, after the dilation, the point changes again to the point $\left(1,-3\right)$(1,−3) on the red transformed curve.