The question is:
Sketch the graph of the the function by transforming the graph of an appropriate function of the form $y = x^n$. Indicate all x and y intercepts on the graph.
I am really trying to understand how to do this freehand before my test on Tuesday. Is there anything else that I should do to make this answer correct? Also, how do I show that the graph shrinks?
Should I make a t-table or is that a waste of time?
Start with the graph of $y=f(x)=x^{3}$.
Then draw $g(x)=(x-1)^{3}=f(x-1)$. This graph can be obtained by translating the graph of $f(x)$ to the right by +1 (Why?). Then draw $h(x)=\frac{1}{2}(x-1)^{3}=\frac{1}{2}g(x)$. Note that the values of $h(x)$ are reduced by a factor of $1/2$ with respect to the equivalent ones of $g(x)$.
Finally draw the graph of $S(x)=\frac{1}{2}(x-1)^{3}+4=\frac{1}{2}g(x)+4=h(x)+4.$
The $x$ intercept is the value of $x$ such that $S(x)=0$, i.e \begin{eqnarray*} \frac{1}{2}(x-1)^{3}+4 &=&0\Leftrightarrow (x-1)^{3}+8=0\Leftrightarrow (x-1)^{3}=-8 \\ &\Leftrightarrow &x-1=\sqrt[3]{-8}=-2\Leftrightarrow x=-2+1=-1 \end{eqnarray*}
The $y$ intercept is $S(0)=\frac{1}{2}(0-1)^{3}+4=\frac{7}{2}$.
ADDED. The two graphs are represented below: $f(x)=x^3$ (black) and $S(x)=\frac{1}{2}(x-1)^3+4$ (red)