Graph the function by using transformations of the graph of $$ y=\frac{1}{x^{2}} $$. Plot all necessary asymptotes. For vertical asymptotes, make sure there are at least two points on each side. $$ h(x)=\frac{1}{x^{2}}+2 $$

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1. Start with the base function  
   $$
   y=\frac{1}{x^{2}}
   $$

2. The function to graph is  
   $$
   h(x)=\frac{1}{x^{2}}+2
   $$  
   This represents a vertical shift of the base graph upward by 2 units.

3. Transformation of key features:  
   - **Vertical asymptote:**  
     The function $$ y=\frac{1}{x^{2}} $$ has a vertical asymptote at $$ x=0 $$. Since the transformation is a vertical shift, the vertical asymptote remains  
     $$
     x=0.
     $$
   - **Horizontal asymptote:**  
     The base function has a horizontal asymptote at $$ y=0 $$. Adding 2 to the function shifts the asymptote upward, so the new horizontal asymptote is  
     $$
     y=2.
     $$

4. Plotting points around the vertical asymptote $$ x=0 $$:  
   Ensure to include at least two points from each side of $$ x=0 $$. For example:
   
   - For $$ x > 0 $$:  
     - At $$ x=0.5 $$:
       $$
       h(0.5)=\frac{1}{(0.5)^2}+2=\frac{1}{0.25}+2=4+2=6.
       $$
     - At $$ x=1 $$:
       $$
       h(1)=\frac{1}{1^2}+2=1+2=3.
       $$
       
   - For $$ x < 0 $$:  
     - At $$ x=-0.5 $$:
       $$
       h(-0.5)=\frac{1}{(-0.5)^2}+2=\frac{1}{0.25}+2=4+2=6.
       $$
     - At $$ x=-1 $$:
       $$
       h(-1)=\frac{1}{1}+2=1+2=3.
       $$

5. Graph summary:  
   - The graph of $$ h(x)=\frac{1}{x^{2}}+2 $$ is the graph of $$ y=\frac{1}{x^{2}} $$ shifted upward by 2 units.  
   - The vertical asymptote remains at $$ x=0 $$ (with points on either side as given above).  
   - The horizontal asymptote is now at $$ y=2 $$.  
   - The curve remains positive for all $$ x \neq 0 $$ and gets very large as $$ x $$ approaches 0 from either side.

This completes the transformation and identifies all the necessary asymptotes and reference points for graphing $$ h(x) $$.
To graph $$ h(x) = \frac{1}{x^{2}} + 2 $$, start with the base function $$ y = \frac{1}{x^{2}} $$ and shift it upward by 2 units. The vertical asymptote remains at $$ x = 0 $$, and the new horizontal asymptote is $$ y = 2 $$. Plot points around $$ x = 0 $$ to show the behavior of the function on both sides.

Graph the function by using transformations of the graph of \( y=\frac{1}{x^{2}} \). Plot all necessary asymptotes. For vertical asymptotes, make sure there are at least two points on each side. \[ h(x)=\frac{1}{x^{2}}+2 \]

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