Question

use transformations of ( f(x)=\frac{1}{x^{2}} ) to graph ( g(x)=\frac{1}{(x + 3)^{2}}+6 ).

Explanation:

Step1: Horizontal shift

For the function \(y = f(x + h)\), if \(h>0\), the graph of \(y = f(x)\) is shifted \(h\) units to the left.
For \(g(x)=\frac{1}{(x + 3)^{2}}\), compared with \(f(x)=\frac{1}{x^{2}}\), \(h = 3\). So the graph of \(y=\frac{1}{x^{2}}\) is shifted \(3\) units to the left to get \(y=\frac{1}{(x + 3)^{2}}\).

Step2: Vertical shift

For the function \(y=f(x)+k\), if \(k>0\), the graph of \(y = f(x)\) is shifted \(k\) units up.
For \(g(x)=\frac{1}{(x + 3)^{2}}+6\), compared with \(y=\frac{1}{(x + 3)^{2}}\), \(k = 6\). So the graph of \(y=\frac{1}{(x + 3)^{2}}\) is shifted \(6\) units up to get \(y=\frac{1}{(x + 3)^{2}}+6\).

Answer:

First, shift the graph of \(y = \frac{1}{x^{2}}\) \(3\) units to the left to obtain the graph of \(y=\frac{1}{(x + 3)^{2}}\). Then, shift the graph of \(y=\frac{1}{(x + 3)^{2}}\) \(6\) units up to get the graph of \(y=\frac{1}{(x + 3)^{2}}+6\).