NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Transformations of Rational Functions

Lesson

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`−2`x`+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`=`x``x`+7 translated $2$2 units to the right. - The rational function $y=\frac{2x^3}{x^2+5}+3$
`y`=2`x`3`x`2+5+3 is similarly the function $y=\frac{2x^3}{x^2+5}$`y`=2`x`3`x`2+5 translated upwards by $3$3 units. - The rational function $y=\frac{-12x}{x+1}$
`y`=−12`x``x`+1 is the function $y=\frac{x}{x+1}$`y`=`x``x`+1 reflected in the $x$`x`axis and dilated by a factor of $12$12.

Find the equation of the transformed function created when the rational function $y=\frac{x^2}{x-1}$`y`=`x`2`x`−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`=`x`2`x`−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)2`x`−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.

Form and use trigonometric, polynomial, and other non-linear equations

Apply trigonometric methods in solving problems