Obtaining a function by using transformations

In response of the question in Yahoo Answers https://goo.gl/jxqXtR



We can obtain the graph of a function by making transformations from an elemental one. We have for instance function $f(x)=x^3$

Regarding transformations, we need to take these properties into consideration:

Taking that in mind, lets transform $f(x)=x^3$ algebraically by given information:

$$\begin{array}{lcl} \text{Horizontal shift right: } 4 & \Longrightarrow & g(x)=(x-4)^3 \\ \text{Horizontal shrink: } \dfrac23 & \Longrightarrow & g(x)=\left(\dfrac23x-4\right)^3 \\ \text{Vertical stretch by } 2 & \Longrightarrow & g(x)=2\cdot\left(\dfrac23x-4\right)^3 \\ \text{Reflection over } x & \Longrightarrow & g(x)=-2\cdot\left(\dfrac23x-4\right)^3 \\ \text{Vertical shift down: } 9 & \Longrightarrow & g(x)=-2\cdot\left(\dfrac23x-4\right)^3-9 \end{array}$$

And expanding the last expresion, we get 

$$\begin{align} g(x) & = -\dfrac{1}{27}\cdot\left(16 x^3 - 288 x^2 + 1728 x -3213\right) \\ & \\ & = -\dfrac{16}{27}x^3 + \dfrac{32}{3}x^2 - 64 x + 119\end{align}$$

Or the compact form

$$g(x)=-\dfrac{16}{27} (x - 6)^3 - 9$$