# How do you write the equation of a transformed graph?

May 27, 2015

Let's consider a simple example: $\textcolor{red}{f \left(x\right) = {x}^{2}}$ is the parent function, represented by the red curve.
The graph of the transformed function $\textcolor{b l u e}{g \left(x\right)}$ is colored in blue. Our goal is to find out the equation of the function $\textcolor{b l u e}{g}$

We notice that the transformed function $\textcolor{b l u e}{g}$ is the outcome of several successive transformations applied to the parent function $\textcolor{red}{f}$:

1. Reflection of $f$ across the $O x$ axis : this gives us the first transformed function $\textcolor{g r e e n}{{f}_{1} \left(x\right) = - {x}^{2}}$

2. Translation of ${f}_{1}$ by 5 units to the right across the $O x$ axis: this gives us the second transformed function $\textcolor{\in \mathrm{di} g o}{{f}_{2} \left(x\right) = - {\left(x - 5\right)}^{2}}$

3. Translation of ${f}_{2}$ by 4 units up across the $O y$ axis : this gives us the transformed function $g \left(x\right) = - {\left(x - 5\right)}^{2} + 4 = {x}^{2} + 10 x - 25 + 4$ $\to$ $\textcolor{b l u e}{g \left(x\right) = - {x}^{2} + 10 x - 21}$, that is the formula we were looking for.

Please find below the graphs of the successive transformations, represented in the corresponding colors (red, green, indigo, blue):